A192849 Molecular topological indices of the triangular graphs.

0, 0, 24, 240, 1080, 3360, 8400, 18144, 35280, 63360, 106920, 171600, 264264, 393120, 567840, 799680, 1101600, 1488384, 1976760, 2585520, 3335640, 4250400, 5355504, 6679200, 8252400, 10108800, 12285000, 14820624, 17758440, 21144480

Offset

1,3

Comments

Triangular graphs are defined for n>=2; extended to n=1 using closed form.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

G. D. Birkhoff, A determinant formula for the number of ways of coloring a map, Ann. Math., 14:42-4. See 2nd polynomial p. 5.

Eric Weisstein's World of Mathematics, Molecular Topological Index.

Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Formula

a(n) = n*(n^2 - 1)*(n-2)^2.

a(n) = 24*A027800(n-3).

G.f.: 24*x^3*(4*x+1)/(x-1)^6. - Colin Barker, Aug 07 2012

a(n) = A245334(n+1,4), n > 2. - Reinhard Zumkeller, Aug 31 2014

E.g.f.: x^3*(4 + 6*x + x^2)*exp(x). - G. C. Greubel, Jan 05 2019

From Amiram Eldar, May 14 2022: (Start)

Sum_{n>=3} 1/a(n) = Pi^2/36 - 49/216.

Sum_{n>=3} (-1)^(n+1)/a(n) = Pi^2/72 - 10*log(2)/9 + 145/216. (End)

Maple

[n*(n^2-1)*(n-2)^2$n=1..40]; # Muniru A Asiru, Jan 05 2019

Mathematica

Table[n*(n^2-1)*(n-2)^2, {n, 1, 40}] (* G. C. Greubel, Jan 05 2019 *)

Prog

(Haskell)
a192849 n = if n < 3 then 0 else a245334 (n + 1) 4
-- Reinhard Zumkeller, Aug 31 2014
(PARI) vector(40, n, n*(n^2 -1)*(n-2)^2) \\ G. C. Greubel, Jan 05 2019
(Magma) [n*(n^2 -1)*(n-2)^2: n in [1..40]]; // G. C. Greubel, Jan 05 2019
(Sage) [n*(n^2 -1)*(n-2)^2 for n in (1..40)] # G. C. Greubel, Jan 05 2019
(GAP) List([1..40], n -> n*(n^2 -1)*(n-2)^2); # G. C. Greubel, Jan 05 2019

Crossrefs

Cf. A027800, A245334.

Sequence in context: A052663 A192491 A167548 * A253285 A052796 A056269

Adjacent sequences: A192846 A192847 A192848 * A192850 A192851 A192852

Keyword

nonn,easy

Author

Eric W. Weisstein, Jul 11 2011

Status

approved