A048920 Indices of heptagonal numbers (A000566) which are also 9-gonal.

1, 104, 14725, 2090804, 296879401, 42154784096, 5985682462189, 849924754846700, 120683329505769169, 17136182865064375256, 2433217283509635517141, 345499718075503179058724

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• easy

Comments

As n increases, this sequence is approximately geometric with common ratio r = lim_{n->infinity} a(n)/a(n-1) = (6 + sqrt(35))^2 = 71 + 12*sqrt(35). - Ant King, Dec 31 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, Nonagonal Heptagonal Number.
• Index entries for linear recurrences with constant coefficients, signature (143,-143,1).

Crossrefs

Cf. A000566, A048919, A048921.

Programs

Magma

I:=[1, 104, 14725]; [n le 3 select I[n] else 143*Self(n-1)-143*Self(n-2)+Self(n-3): n in [1..30]]; // Vincenzo Librandi, Dec 21 2011

Mathematica

LinearRecurrence[{143,-143,1},{1,104,14725},30] (* Vincenzo Librandi, Dec 21 2011 *)

Maxima

makelist(expand((42+(-21+5*sqrt(35))*(6+sqrt(35))^(2*n-1)-(21+5*sqrt(35))*(6-sqrt(35))^(2*n-1))/140), n, 1, 12); /* Bruno Berselli, Dec 20 2011 */

Formula

From Bruno Berselli, Dec 20 2011: (Start)

G.f.: x*(1 - 39*x - 4*x^2)/((1-x)*(1 - 142*x + x^2)).

a(n) = (42 + (-21+5r)*(6+r)^(2n-1) - (21+5r)*(6-r)^(2n-1))/140, where r=sqrt(35). (End)

From Ant King, Dec 31 2011: (Start)

a(n) = 142*a(n-1) - a(n-2) - 42.

a(n) = ceiling(1/140*(49+9*sqrt(35))*(6+sqrt(35))^(2*n-2)).

(End)