A048923 Indices of octagonal numbers which are also 9-gonal.

1, 459, 309141, 208360351, 140434567209, 94652689938291, 63795772583840701, 42998256068818693959, 28980760794611215887441, 19532989777311890689441051, 13165206129147419713467380709

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) = (sqrt(6) + sqrt(7))^4 = 337 + 52*sqrt(42). - Ant King, Jan 03 2012

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Eric Weisstein's World of Mathematics, Nonagonal Octagonal Number.
• Index entries for linear recurrences with constant coefficients, signature (675,-675,1).

Crossrefs

Cf. A048922, A048924.

Programs

Magma

I:=[1, 459, 309141]; [n le 3 select I[n] else 675*Self(n-1)-675*Self(n-2)+Self(n-3): n in [1..15]]; // Vincenzo Librandi, Dec 24 2011

Mathematica

LinearRecurrence[{675,-675,1},{1,459,309141},30] (* Vincenzo Librandi, Dec 24 2011 *)

Formula

G.f.: x*(-1 + 216*x + 9*x^2) / ( (x-1)*(x^2 - 674*x + 1) ). - R. J. Mathar, Dec 21 2011

From Ant King, Jan 03 2012: (Start)

a(n) = 674*a(n-1) - a(n-2) - 224.

a(n) = (1/168)*((7*sqrt(6) + 2*sqrt(7))*(sqrt(6) + sqrt(7))^(4*n-3) + (7*sqrt(6) - 2*sqrt(7))*(sqrt(6) - sqrt(7))^(4*n-3) + 56).

a(n) = ceiling((1/168)*(7*sqrt(6) + 2*sqrt(7))*(sqrt(6) + sqrt(7))^(4*n-3)). (End)