A048902 Indices of heptagonal numbers (A000566) which are also hexagonal.

1, 221, 71065, 22882613, 7368130225, 2372515049741, 763942477886281, 245987105364332645, 79207083984837225313, 25504435056012222218045, 8212348880951950716985081, 2644350835231472118646977941

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) = (2 + sqrt(5))^4 = 161 + 72*sqrt(5). - Ant King, Dec 26 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, Heptagonal hexagonal number.
• Index entries for linear recurrences with constant coefficients, signature (323,-323,1).

Crossrefs

Cf. A048901, A048903.

Programs

Magma

I:=[1, 221, 71065]; [n le 3 select I[n] else 323*Self(n-1)-323*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Dec 28 2011

Mathematica

LinearRecurrence[{323, -323, 1}, {1, 221, 71065}, 12]; (* Ant King, Dec 26 2011 *)

Formula

G.f.: -x*(1 - 102*x + 5*x^2) / ( (x-1)*(x^2 - 322*x + 1) ). - R. J. Mathar, Dec 21 2011

From Ant King, Dec 26 2011: (Start)

a(n) = 322*a(n-1) - a(n-2) - 96.

a(n) = (1/20)*((sqrt(5)+1)*(sqrt(5)+2)^(4*n-3) + (sqrt(5)-1)*(sqrt(5)-2)^(4*n-3) + 6).

a(n) = ceiling((1/20)*(sqrt(5)+1)*(sqrt(5)+2)^(4*n-3)).

(End)