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A000602.html
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A000602.html
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
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Greetings from <a href="http://www.research.att.com/%7Enjas/sequences/Seis.html">The On-Line Encyclopedia of Integer Sequences</a>!
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<td>Displaying 1-3 of 3 results found.</td>
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<td align="left" valign="top" width="100"><a href="http://www.research.att.com/%7Enjas/sequences/A000602" title="Number of n-node unrooted quartic trees; number of n-carbon alkanes C(n)H(2n+2) ignoring stereoisomers.">A000602</a></td>
<td width="5"></td>
<td align="left" valign="top">
Number of n-node unrooted quartic trees; number of n-carbon alkanes C(n)H(2n+2) ignoring stereoisomers. <br><font size="-1">(Formerly M0718 N0267)</font>
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<font size="-2">+20<br>26</font>
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<tbody><tr><td width="20"></td><td width="710"><tt>1, 1, 1, 1, 2, <b style="color: black; background-color: rgb(255, 255, 102);">3, 5, 9, 18, 35, 75</b>,
159, 355, 802, 1858, 4347, 10359, 24894, 60523, 148284, 366319, 910726,
2278658, 5731580, 14490245, 36797588, 93839412, 240215803, 617105614,
1590507121, 4111846763, 10660307791, 27711253769</tt>
<font size="-1">(<a href="http://www.research.att.com/%7Enjas/sequences/table?a=602&fmt=4">list</a>; <a href="http://www.research.att.com/%7Enjas/sequences/table?a=602&fmt=5">graph</a>; <a href="http://www.research.att.com/%7Enjas/sequences/table?a=602&fmt=6">listen</a>)</font>
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<font size="-2">OFFSET</font>
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<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>0,5</tt></p>
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<font size="-2">COMMENT</font>
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<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
Trees are unrooted, nodes are unlabeled. Every node has degree <= 4.
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
Ignoring stereoisomers means that the children of a node are unordered.
They can be permuted in any way and it is still the same tree. See <a href="http://www.research.att.com/%7Enjas/sequences/A000628" title="Number of n-node unrooted steric quartic trees; number of n-carbon alkanes C(n)H(2n+2) taking stereoisomers into account.">A000628</a> for the analogous sequence with stereoisomers counted.
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
In alkanes every carbon has valence exactly 4 and every hydrogen has
valence exactly 1. But the trees considered here are just the carbon
"skeletons" (with all the hydrogens stripped off) so now each carbon
bonds to 1 to 4 other carbons. The degree of each node is then <= 4.
</tt></p>
</td></tr>
<tr><td width="20"></td><td valign="top" width="100">
<font size="-2">REFERENCES</font>
</td>
<td width="600">
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
M. van Almsick, H. Dolhaine and H. Honig, Efficient algorithms to
enumerate isomers and diamutamers with more than one type of
substituent, J. Chem. Info. and Computer Science, 40 (2000), 956-966. </tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Camb. 1998, p. 290.
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
L. Bytautas and D. J. Klein, Chemical combinatorics for alkane-isomer
enumeration and more, J. Chem. Inf. Comput. Sci., 38 (1998), 1063-1078.
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
A. Cayley, Ueber die analytischen Figuren, welche in der Mathematik
Baeume genannt werden..., Chem. Ber. 8 (1875), 1056-1059. </tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
R. Davies and P. J. Freyd, C_{167}H_{336} is The Smallest Alkane with
More Realizable Isomers than the Observable Universe has Particles,
Journal of Chemical Education, Vol. 66, April 1989, pp. 278-281. </tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
J. L. Faulon, D. Visco and D. Roe, Enumerating Molecules, In: Reviews
in Computational Chemistry Vol. 21, Ed. K. Lipkowitz, Wiley-VCH, 2005. </tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
Handbook of Combinatorics, North-Holland '95, p. 1963.
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
F. Harary and R. Z. Norman, Dissimilarity characteristic theorems for
graphs, Proc. Amer. Math. Soc. 11 (1960), 332-334. </tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
H. R. Henze and C. M. Blair, The number of isomeric hydrocarbons of the
methane series, J. Amer. Chem. Soc., 53 (1931), 3077-3085. </tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
H. R. Henze and C. M. Blair, The number of structurally isomeric
hydrocarbons of the ethylene series, J. Amer. Chem. Soc., 55 (1933),
680-686. </tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
M. D. Jackson and T. I. Bieber, Applications of degree distribution, 2:
construction and enumeration of isomers in the alkane series, J. Chem.
Info. and Computer Science, 33 (1993), 701-708. </tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
E. V. Konstantinova and M. V. Vidyuk, Discriminating tests of
information and toplogical indices; animals and trees; J. Chem. Inf.
Comput. Sci., 43 (2003), 1860-1871. </tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
J. Lederberg et al., Applications of artificial intelligence for
chemical systems, I: The number of possible organic compounds. Acyclic
structures containing C, H, O and N, J. Amer. Chem. Soc., 91 (1969),
2973-2097. </tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
P. Leroux and B. Miloudi, ``G\'{e}n\'{e}ralisations de la formule
d'Otter,'' Ann. Sci. Math. Qu\'{e}bec, Vol. 16, No. 1, pp. 53-80, 1992.
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
L. M. Masinter, Applications of artificial intelligence for chemical
systems, XX, Exhaustive generation of cyclic and acyclic isomers, J.
Amer. Chem. Soc., 96 (1974), 7702-7714. </tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
W. R. Mueller et al., Molecular topological index, J. Chem. Inf. Comput. Sci., 30 (1990),160-163.
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
D. Perry, The number of structural isomers ..., J. Amer. Chem. Soc. 54
(1932), 2918-2920. [Gives a(60) correctly - compare first link below] </tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
M. Petkovsek and T. Pisanski, Counting disconnected structures:
chemical trees, fullerenes, I-graphs and others, Croatica Chem. Acta,
78 (2005), 563-567. </tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
G. Polya, Algebraische Berechnung der Anzahl der Isomeren einiger
organischer Verbindungen, Zeit. f. Kristall., 93 (1936), 415-443. </tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
R. C. Read, The Enumeration of Acyclic Chemical Compounds, pp. 25-61 of
A. T. Balaban, ed., Chemical Applications of Graph Theory, Ac. Press,
1976; see p. 28. </tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
R. W. Robinson, F. Harary and A. T. Balaban, Numbers of chiral and
achiral alkanes..., Tetrahedron 32 (1976), 355-361. </tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
ten Hoor, Education in Chemistry (details needed).
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
N. Trinajsti, Z. Jerievi, J. V. Knop, W. R. Muller and K. Szymanski,
COMPUTER GENERATION OF ISOMERIC STRUCTURES, Pure & Appl. Chem.,
Vol. 55, No. 2, pp. 379-39O, 1983. </tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
S. Wagner, Graph-theoretical enumeration and digital expansions: an
analytic approach, Dissertation, Fakult. f. Tech. Math. u. Tech.
Physik, Tech. Univ. Graz, Austria, Feb., 2006. </tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
J. L. Gross and J. Yellen, eds., Handbook of Graph Theory, CRC Press, 2004; p. 529.
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
J. B. Hendrickson and C. A. Parks, "Generation and Enumeration of
Carbon skeletons", J. Chem. Inf. Comput. Sci, vol. 31 (1991) pp.
101-107. See Table 2, column 2 on page 103. </tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
A. T. Balaban, J. W. Kennedy and L. V. Quintas, The number of alkanes
having n carbons and a longest chain of length d, J. Chem. Education,
65 (No. 4, 1988), 304-313. </tt></p>
</td></tr>
<tr><td width="20"></td><td valign="top" width="100">
<font size="-2">LINKS</font>
</td>
<td width="600">
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
N. J. A. Sloane, <a href="http://www.research.att.com/%7Enjas/sequences/b000602.txt">Table of n, a(n) for n = 0..60</a>
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
R. Aringhieri, P. Hansen and F. Malucelli, <a href="http://citeseer.ist.psu.edu/aringhieri99chemical.html">Chemical Tree Enumeration Algorithms</a>, Report TR-99-09, Dept. Informatica, Univ. Pisa, 1999.
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
H. Bottomley, <a href="http://www.research.att.com/%7Enjas/sequences/a602.gif">Illustration of initial terms of A000022, A000200, A000602</a>
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
Michael A. Kappler, <a href="http://www.daylight.com/meetings/emug04/Kappler/GenSmi.html">GENSMI: Exhaustive Enumeration of Simple Graphs.</a> Daylight CIS, Inc., EuroMUG '04;4-5 Nov, 2004.
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
E. M. Rains and N. J. A. Sloane, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">On Cayley's Enumeration of Alkanes (or 4-Valent Trees).</a>, J. Integer Sequences, Vol. 2 (1999), Article 99.1.1.
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
N. J. A. Sloane, <a href="http://www.research.att.com/%7Enjas/sequences/a000602.txt">Maple program and first 60 terms for A000022, A000200, A000598, A000602, A000678</a>
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
<a href="http://www.research.att.com/%7Enjas/sequences/Sindx_Tra.html#trees">Index entries for sequences related to trees</a>
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
<a href="http://www.research.att.com/%7Enjas/sequences/Sindx_Cor.html#core">Index entries for "core" sequences</a>
</tt></p>
</td></tr>
<tr><td width="20"></td><td valign="top" width="100">
<font size="-2">FORMULA</font>
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<td width="600">
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
a(n) = <a href="http://www.research.att.com/%7Enjas/sequences/A010372" title="Number of unrooted quartic trees with n (unlabeled) nodes and possessing a centroid; number of n-carbon alkanes C(n)H(2n +2) wit">A010372</a>(n) + <a href="http://www.research.att.com/%7Enjas/sequences/A010373" title="Number of unrooted quartic trees with 2n (unlabeled) nodes and possessing a bicentroid; number of 2n-carbon alkanes C(2n)H(4n+2)">A010373</a>(n/2) for n even, a(n) = <a href="http://www.research.att.com/%7Enjas/sequences/A010372" title="Number of unrooted quartic trees with n (unlabeled) nodes and possessing a centroid; number of n-carbon alkanes C(n)H(2n +2) wit">A010372</a>(n) for n odd.
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
Also equals <a href="http://www.research.att.com/%7Enjas/sequences/A000022" title="Number of centered hydrocarbons with n atoms.">A000022</a> + <a href="http://www.research.att.com/%7Enjas/sequences/A000200" title="Number of bicentered hydrocarbons with n atoms.">A000200</a> (n>0), both of which have known generating functions. Also g.f. = <a href="http://www.research.att.com/%7Enjas/sequences/A000678" title="Number of carbon (rooted) trees with n carbon atoms = unordered 4-tuples of ternary trees.">A000678</a>(x)-<a href="http://www.research.att.com/%7Enjas/sequences/A000599" title="Number of secondary alcohols (alkanols or alkyl alcohols C_n H_{2n+1} OH) with n carbon atoms.">A000599</a>(x)+<a href="http://www.research.att.com/%7Enjas/sequences/A000598" title="Number of rooted ternary trees with n nodes; number of n-carbon alkyl radicals C(n)H(2n+1) ignoring stereoisomers.">A000598</a>(x^2) = (x+x^2+2x^3...)-(x^2+x^3+3x^4...)+(1+x^2+x^4+...) = 1+x+x^2+x^3+2x^4+3x^5...
</tt></p>
</td></tr>
<tr><td width="20"></td><td valign="top" width="100">
<font size="-2">EXAMPLE</font>
</td>
<td width="600">
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
a(6)=5 because hexane has five isomers: n-hexane, 2-methylhexane,
3-methylhexane, 2,2-dimethylhexane, 2,3-dimethylhexane. - Michael Lugo
(mtlugo(AT)mit.edu), Mar 15 2003 </tt></p>
</td></tr>
<tr><td width="20"></td><td valign="top" width="100">
<font size="-2">MAPLE</font>
</td>
<td width="600">
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
A000602 := proc(n) if n=0 then RETURN(1) else A000022(n)+A000200(n); fi; end;
</tt></p>
</td></tr>
<tr><td width="20"></td><td valign="top" width="100">
<font size="-2">CROSSREFS</font>
</td>
<td width="600">
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
Cf. <a href="http://www.research.att.com/%7Enjas/sequences/A000598" title="Number of rooted ternary trees with n nodes; number of n-carbon alkyl radicals C(n)H(2n+1) ignoring stereoisomers.">A000598</a>, <a href="http://www.research.att.com/%7Enjas/sequences/A000625" title="Number of n-node steric rooted ternary trees; number of n carbon alkyl radicals C(n)H(2n+1) taking stereoisomers into account">A000625</a>, <a href="http://www.research.att.com/%7Enjas/sequences/A000628" title="Number of n-node unrooted steric quartic trees; number of n-carbon alkanes C(n)H(2n+2) taking stereoisomers into account.">A000628</a>, <a href="http://www.research.att.com/%7Enjas/sequences/A067608" title="Number of structural alkanes with combinatorial diameter n.">A067608</a>-<a href="http://www.research.att.com/%7Enjas/sequences/A067610" title="Number of stereo alkanes not containing a certain forbidden substructure) with combinatorial diameter n.">A067610</a>.
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
Adjacent sequences: <a href="http://www.research.att.com/%7Enjas/sequences/A000599" title="Number of secondary alcohols (alkanols or alkyl alcohols C_n H_{2n+1} OH) with n carbon atoms.">A000599</a> <a href="http://www.research.att.com/%7Enjas/sequences/A000600" title="Number of tertiary alcohols (alkanols or alkyl alcohols C_n H_{2n+1} OH) with n carbon atoms.">A000600</a> <a href="http://www.research.att.com/%7Enjas/sequences/A000601" title="Expansion of 1/((1-x)^2*(1-x^2)*(1-x^3)).">A000601</a> this_sequence <a href="http://www.research.att.com/%7Enjas/sequences/A000603" title="Number of nonnegative solutions to x^2 + y^2 <= n^2.">A000603</a> <a href="http://www.research.att.com/%7Enjas/sequences/A000604" title="Number of nonnegative solutions to x^2 + y^2 + z^2 <= n^2.">A000604</a> <a href="http://www.research.att.com/%7Enjas/sequences/A000605" title="Number of points of norm <= n in cubic lattice.">A000605</a>
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
Sequence in context: <a href="http://www.research.att.com/%7Enjas/sequences/A047031" title="Sum{T(i,n-i): i=0,1,...,n}, array T as in A047030.">A047031</a> <a href="http://www.research.att.com/%7Enjas/sequences/A056766" title="Smallest member of A056757 (numbers for which cube of number of divisors exceeds the number) between 2^(n-1) and 2^n.">A056766</a> <a href="http://www.research.att.com/%7Enjas/sequences/A080091" title="Duplicate of A000602.">A080091</a> this_sequence <a href="http://www.research.att.com/%7Enjas/sequences/A034790" title="Erroneous version of A000602.">A034790</a> <a href="http://www.research.att.com/%7Enjas/sequences/A047121" title="Sum{T(i,n-i): i=0,1,...,n}, array T as in A047120.">A047121</a> <a href="http://www.research.att.com/%7Enjas/sequences/A096753" title="Antidiagonal sums of table A096751.">A096753</a>
</tt></p>
</td></tr>
<tr><td width="20"></td><td valign="top" width="100">
<font size="-2">KEYWORD</font>
</td>
<td width="600">
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
nonn,easy,core,nice
</tt></p>
</td></tr>
<tr><td width="20"></td><td valign="top" width="100">
<font size="-2">AUTHOR</font>
</td>
<td width="600">
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
njas
</tt></p>
</td></tr>
<tr><td width="20"></td><td valign="top" width="100">
<font size="-2">EXTENSIONS</font>
</td>
<td width="600">
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
Additional comments from Steve Strand (snstrand(AT)comcast.net), Aug 20, 2003.
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
Kappler reference from Jonathan Vos Post (jvospost2(AT)yahoo.com), Dec 15 2005
</tt></p>
</td></tr>
</tbody></table>
</td></tr>
<tr height="10"><td></td></tr>
<tr bgcolor="#76767f" height="1"><td>
</td></tr><tr bgcolor="#eeeeff"><td valign="top">
<table border="0" cellpadding="0" cellspacing="0" width="100%">
<tbody><tr>
<td align="left" valign="top" width="100"><a href="http://www.research.att.com/%7Enjas/sequences/A034790" title="Erroneous version of A000602.">A034790</a></td>
<td width="5"></td>
<td align="left" valign="top">
Erroneous version of <a href="http://www.research.att.com/%7Enjas/sequences/A000602" title="Number of n-node unrooted quartic trees; number of n-carbon alkanes C(n)H(2n+2) ignoring stereoisomers.">A000602</a>. <br><font size="-1">(Formerly N0268)</font>
</td>
<td width="2"></td>
<td align="right" valign="top">
<font size="-2">+20<br>1</font>
</td>
</tr>
</tbody></table>
</td></tr><tr><td>
<table cellborder="0" cellpadding="2" cellspacing="0">
<tbody><tr><td width="20"></td><td width="710"><tt>1, 1, 1, 2, <b style="color: black; background-color: rgb(255, 255, 102);">3, 5, 9, 18, 35, 75</b>, 159, 357, 799</tt>
<font size="-1">(<a href="http://www.research.att.com/%7Enjas/sequences/table?a=34790&fmt=4">list</a>; <a href="http://www.research.att.com/%7Enjas/sequences/table?a=34790&fmt=5">graph</a>; <a href="http://www.research.att.com/%7Enjas/sequences/table?a=34790&fmt=6">listen</a>)</font>
</td></tr></tbody></table>
</td></tr>
<tr><td valign="top">
<table cellborder="0" cellpadding="2" cellspacing="0">
<tbody><tr><td width="20"></td><td valign="top" width="100">
<font size="-2">OFFSET</font>
</td>
<td width="600">
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>1,4</tt></p>
</td>
<td width="20"></td>
</tr><tr><td width="20"></td><td valign="top" width="100">
<font size="-2">CROSSREFS</font>
</td>
<td width="600">
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
Adjacent sequences: <a href="http://www.research.att.com/%7Enjas/sequences/A034787" title="One fifth of sextic factorial numbers.">A034787</a> <a href="http://www.research.att.com/%7Enjas/sequences/A034788" title="One sixth of sextic factorial numbers.">A034788</a> <a href="http://www.research.att.com/%7Enjas/sequences/A034789" title="Related to sextic factorial numbers A008542.">A034789</a> this_sequence <a href="http://www.research.att.com/%7Enjas/sequences/A034791" title="a(n) is square mod a(i), i < n.">A034791</a> <a href="http://www.research.att.com/%7Enjas/sequences/A034792" title="a(n) triangular, coprime to a(i), i < n.">A034792</a> <a href="http://www.research.att.com/%7Enjas/sequences/A034793" title="a(n) is square mod a(i) for all i < n.">A034793</a>
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
Sequence in context: <a href="http://www.research.att.com/%7Enjas/sequences/A056766" title="Smallest member of A056757 (numbers for which cube of number of divisors exceeds the number) between 2^(n-1) and 2^n.">A056766</a> <a href="http://www.research.att.com/%7Enjas/sequences/A080091" title="Duplicate of A000602.">A080091</a> <a href="http://www.research.att.com/%7Enjas/sequences/A000602" title="Number of n-node unrooted quartic trees; number of n-carbon alkanes C(n)H(2n+2) ignoring stereoisomers.">A000602</a> this_sequence <a href="http://www.research.att.com/%7Enjas/sequences/A047121" title="Sum{T(i,n-i): i=0,1,...,n}, array T as in A047120.">A047121</a> <a href="http://www.research.att.com/%7Enjas/sequences/A096753" title="Antidiagonal sums of table A096751.">A096753</a> <a href="http://www.research.att.com/%7Enjas/sequences/A022862" title="a(n) = [ a(n-1)/a(1) ] + [ a(n-1)/a(2) ] + ... + [ a(n-1)/a(n-1) ] for n >= 3.">A022862</a>
</tt></p>
</td></tr>
<tr><td width="20"></td><td valign="top" width="100">
<font size="-2">KEYWORD</font>
</td>
<td width="600">
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
dead
</tt></p>
</td></tr>
</tbody></table>
</td></tr>
<tr height="10"><td></td></tr>
<tr bgcolor="#76767f" height="1"><td>
</td></tr><tr bgcolor="#eeeeff"><td valign="top">
<table border="0" cellpadding="0" cellspacing="0" width="100%">
<tbody><tr>
<td align="left" valign="top" width="100"><a href="http://www.research.att.com/%7Enjas/sequences/A080091" title="Duplicate of A000602.">A080091</a></td>
<td width="5"></td>
<td align="left" valign="top">
Duplicate of <a href="http://www.research.att.com/%7Enjas/sequences/A000602" title="Number of n-node unrooted quartic trees; number of n-carbon alkanes C(n)H(2n+2) ignoring stereoisomers.">A000602</a>. </td>
<td width="2"></td>
<td align="right" valign="top">
<font size="-2">+20<br>1</font>
</td>
</tr>
</tbody></table>
</td></tr><tr><td>
<table cellborder="0" cellpadding="2" cellspacing="0">
<tbody><tr><td width="20"></td><td width="710"><tt>1, 1, 1, 1, 2, <b style="color: black; background-color: rgb(255, 255, 102);">3, 5, 9, 18, 35, 75</b>, 159, 355, 802, 1858, 4347, 10359, 24894, 60523</tt>
<font size="-1">(<a href="http://www.research.att.com/%7Enjas/sequences/table?a=80091&fmt=4">list</a>; <a href="http://www.research.att.com/%7Enjas/sequences/table?a=80091&fmt=5">graph</a>; <a href="http://www.research.att.com/%7Enjas/sequences/table?a=80091&fmt=6">listen</a>)</font>
</td></tr></tbody></table>
</td></tr>
<tr><td valign="top">
<table cellborder="0" cellpadding="2" cellspacing="0">
<tbody><tr><td width="20"></td><td valign="top" width="100">
<font size="-2">OFFSET</font>
</td>
<td width="600">
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>0,5</tt></p>
</td>
<td width="20"></td>
</tr><tr><td width="20"></td><td valign="top" width="100">
<font size="-2">CROSSREFS</font>
</td>
<td width="600">
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
Adjacent sequences: <a href="http://www.research.att.com/%7Enjas/sequences/A080088" title="Let r+i*s be the sum of the first-quadrant Gaussian primes dividing prime(n); sequence gives r values.">A080088</a> <a href="http://www.research.att.com/%7Enjas/sequences/A080089" title="Let r+i*s be the sum of the first-quadrant Gaussian primes dividing prime(n); sequence gives s values.">A080089</a> <a href="http://www.research.att.com/%7Enjas/sequences/A080090" title="a(n) = first occurrence of n in A080071, or 0 for those n>0 which never occur in A080071.">A080090</a> this_sequence <a href="http://www.research.att.com/%7Enjas/sequences/A080092" title="Table of prime sequences (p-1|2n) appearing in the n-th von Staudt-Clasuen sum.">A080092</a> <a href="http://www.research.att.com/%7Enjas/sequences/A080093" title="Let sum(k>=0, k^n/(2*k+1)!) = (x(n)*e + y(n)/e)/z(n), where x(n) and z(n) are >0, then a(n)=x(n).">A080093</a> <a href="http://www.research.att.com/%7Enjas/sequences/A080094" title="Let sum(k>=0, k^n/(2*k+1)!) = (x(n)*e + y(n)/e)/z(n), where x(n) and z(n) are >0, then a(n)=y(n).">A080094</a>
</tt></p>
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
Sequence in context: <a href="http://www.research.att.com/%7Enjas/sequences/A047021" title="Sum{T(i,n-i): i=0,1,...,n}, array T given by A047020.">A047021</a> <a href="http://www.research.att.com/%7Enjas/sequences/A047031" title="Sum{T(i,n-i): i=0,1,...,n}, array T as in A047030.">A047031</a> <a href="http://www.research.att.com/%7Enjas/sequences/A056766" title="Smallest member of A056757 (numbers for which cube of number of divisors exceeds the number) between 2^(n-1) and 2^n.">A056766</a> this_sequence <a href="http://www.research.att.com/%7Enjas/sequences/A000602" title="Number of n-node unrooted quartic trees; number of n-carbon alkanes C(n)H(2n+2) ignoring stereoisomers.">A000602</a> <a href="http://www.research.att.com/%7Enjas/sequences/A034790" title="Erroneous version of A000602.">A034790</a> <a href="http://www.research.att.com/%7Enjas/sequences/A047121" title="Sum{T(i,n-i): i=0,1,...,n}, array T as in A047120.">A047121</a>
</tt></p>
</td></tr>
<tr><td width="20"></td><td valign="top" width="100">
<font size="-2">KEYWORD</font>
</td>
<td width="600">
<p style="text-indent: -1em; margin-left: 1em; margin-top: 0pt; margin-bottom: 0pt;"><tt>
dead
</tt></p>
</td></tr>
</tbody></table>
</td></tr>
<tr height="10"><td></td></tr>
</tbody></table>
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<font size="-1">Last modified August 12 14:42 EDT 2007. Contains 131554 sequences.
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</p><center> <font size="-1"> <a href="http://www.research.att.com/">AT&T Labs Research</a> </font> </center> <p>
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