A048922 Indices of 9-gonal numbers which are also octagonal.

1, 425, 286209, 192904201, 130017145025, 87631362842409, 59063408538638401, 39808649723679439625, 26830970850351403668609, 18084034544487122393202601, 12188612452013470141614884225

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 Numbers.
• Index entries for linear recurrences with constant coefficients, signature (675,-675,1).

Crossrefs

Cf. A048923, A048924.

Programs

Magma

I:=[1, 425, 286209]; [n le 3 select I[n] else 675*Self(n-1)-675*Self(n-2)+1*Self(n-3): n in [1..20]]; // Vincenzo Librandi, Dec 23 2011

Mathematica

LinearRecurrence[{675,-675,1},{1,425,286209},30] (* Vincenzo Librandi, Dec 23 2011 *)
Join[{1},Transpose[NestList[{Last[#],674Last[#]-First[#]-240}&, {1,425}, 10]][[2]]] (* Harvey P. Dale, Feb 05 2012 *)

Formula

G.f.: -x*(1 - 250*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) - 240.

a(n) = (1/84)*((sqrt(6) + 3*sqrt(7))*(sqrt(6) + sqrt(7))^(4*n-3) + (sqrt(6) - 3*sqrt(7))*(sqrt(6) - sqrt(7))^(4*n-3) + 30).

a(n) = ceiling((1/84)*(sqrt(6) + 3*sqrt(7))*(sqrt(6) + sqrt(7))^(4*n-3)). (End)