A000602 -id:A000602 - cp's OEIS frontend

A006001 Number of paraffins.

1, 4, 10, 22, 43, 76, 124, 190, 277, 388, 526, 694, 895, 1132, 1408, 1726, 2089, 2500, 2962, 3478, 4051, 4684, 5380, 6142, 6973, 7876, 8854, 9910, 11047, 12268, 13576, 14974, 16465, 18052, 19738, 21526

Offset: 0

N. J. A. Sloane, Simon Plouffe

nonn

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

S. M. Losanitsch, Die Isomerie-Arten bei den Homologen der Paraffin-Reihe, Chem. Ber. 30 (1897), 1917-1926. (Annotated scanned copy)
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (4, -6, 4, -1).

A006001:=(1+2*z**3)/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation

G.f.: (1 + 2 x^3) / (1 - x)^4.

a(n) = 1 + 5*n/2 + n^3/2. - Enrique Pérez Herrero, Mar 19 2012

A289159 Number of unlabeled connected simple graphs with n nodes of degree 4 or less, except trees and 4-regular graphs.

Original entry on oeis.org

0, 0, 1, 4, 17, 72, 342, 1905, 12156, 89268, 738911, 6798758, 68520038, 748502910, 8788173335, 110193643134, 1468070949946, 20694573672876, 307578336064984, 4805384561143819, 78708210390090371, 1348365584611572634, 24108682771962822336, 449043803118075490633

Offset: 1

Natan Arie Consigli, Jul 07 2017

nonn

Also number of saturated cyclic hydrocarbons (cycloalkanes and alkyl cycloalkanes) with n carbon atoms satisfying the octet rule. Stereoisomerism is not taken into account. Some of the drawn structures may or may not exist due to their strained geometries.

Cf. A000602, A006820, A121941, A287424.

a(n) = A121941(n) - A006820(n) - A000602(n).

a(n) = A287424(n) - A000602(n).

a(0) removed and a(16)-a(24) from Andrew Howroyd, Mar 20 2020

A381216 Number of isomers of C_n H_{2n+2} O_2.

Original entry on oeis.org

1, 2, 5, 11, 28, 69, 179, 463, 1225, 3246, 8697, 23366, 63137, 171051, 465002, 1266831, 3459262, 9462393, 25926939, 71139400, 195451500, 537608802, 1480316960, 4079977874, 11254956840, 31072771980, 85850016944, 237356027117, 656657132953, 1817758531055, 5034725293449

Offset: 0

Erich Friedman, Feb 17 2025

nonn

			a(2)=5 because the following compounds are possible:
     | |       |   |       | |       |     |       |
-O-O-C-C-   -O-C-O-C-   -O-C-C-O-   -C-O-O-C-   -O-C-O-
     | |       |   |       | |       |     |       |
                                                  -C-
                                                   |

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Cf. A000598, A000599, A000600, A000602, A000635, A000636.

\\ here R(n) gives g.f. of A000598.
R(n)={my(g=O(x)); for(n=0, n, g = 1 + x*(g^3/6 + subst(g, x, x^2)*g/2 + subst(g, x, x^3)/3) + O(x*x^n)); g}
seq(n)={my(A=O(x*x^n), p=R(n), p2=subst(p,x,x^2) + A, q=(p^2+p2)/2, q2=subst(q,x,x^2) + A); Vec((p^2/(1 - x^2*q^2) + p2/(1 - x^2*q2))*(1 + x*q)/2)} \\ Andrew Howroyd, Feb 17 2025

a(10) onwards from Andrew Howroyd, Feb 17 2025

A121317 Minimum molecular topological index for alkane trees with n vertices.

Original entry on oeis.org

0, 4, 16, 36, 64, 106, 156, 214, 298, 390, 490, 616, 750, 892, 1060, 1236

Offset: 1

Parthasarathy Nambi, Aug 25 2006

nonn

See page 161 of Muller et al. paper for the definition of molecular topological index.

			If n=10 then the minimum molecular topological index is 390 which is the tenth term in the sequence.

W. R. Muller, K. Szymanski, J. V. Knop and N. Trinajstich, "Molecular topological index", J. Chem. Inf. Comput. Sci., vol. 30 (1990), 160-163. See Table 2 on page 162 column 3.

Cf. A000602.

A121318 Molecular topological indices of the path graphs P_n.

Original entry on oeis.org

0, 4, 16, 38, 74, 128, 204, 306, 438, 604, 808, 1054, 1346, 1688, 2084, 2538, 3054, 3636, 4288, 5014, 5818, 6704, 7676, 8738, 9894, 11148, 12504, 13966, 15538, 17224, 19028, 20954, 23006, 25188, 27504, 29958, 32554, 35296, 38188, 41234, 44438

Offset: 1

Parthasarathy Nambi, Aug 25 2006

Maximum molecular topological index for alkane trees with n vertices.

See page 161 of Muller et al. paper for the definition of molecular topological index.

			If n=10 then the maximum molecular topological index is 604 which is the tenth term in the sequence.

W. R. Müller, K. Szymanski, J. V. Knop and N. Trinajstich, Molecular topological index, J. Chem. Inf. Comput. Sci., vol. 30 (1990), pp. 160-163. See Table 2 on page 162 column 4.
Eric Weisstein's World of Mathematics, Molecular Topological Index
Eric Weisstein's World of Mathematics, Path Graph
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000602.

a(1) = 0; a(n) = (2*n^3-3*n^2+13*n-18)/3. - Eric W. Weisstein, Jul 10 2011

G.f.: 2*x^2*(x+1)*(2-2*x+x^2)/(1-x)^4. - Colin Barker, Aug 07 2012

A126886 Triangle read by rows, arising in enumeration of 4-trees by diameter.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 5, 6, 8, 7, 7, 5, 5, 3, 2, 1, 1, 1, 2, 7, 12, 23, 30, 42, 47, 55, 53, 53, 45, 40, 29, 23

Offset: 0

N. J. A. Sloane, Aug 04 2007

See reference for precise definition.

			Triangle begins:
1
1
1 1 1
1 1 2 1 1
1 2 5 6 8 7 7 5 5 3 2 1 1
1 2 7 12 23 30 42 47 55 53 53 45 40 29 23 ...

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 (1988), 304-313.

Cf. A000602 (diagonal sums when rows are displaced by the appropriate amounts), A036437.

cp's OEIS Frontend

A006001 Number of paraffins.

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A289159 Number of unlabeled connected simple graphs with n nodes of degree 4 or less, except trees and 4-regular graphs.

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A381216 Number of isomers of C_n H_{2n+2} O_2.

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PARI

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A121317 Minimum molecular topological index for alkane trees with n vertices.

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A121318 Molecular topological indices of the path graphs P_n.

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A126886 Triangle read by rows, arising in enumeration of 4-trees by diameter.

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