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

1, 459, 309141, 208360351, 140434567209, 94652689938291, 63795772583840701, 42998256068818693959, 28980760794611215887441, 19532989777311890689441051, 13165206129147419713467380709, 8873329398055583574986325156591, 5980610849083334182121069688161401, 4030922838952769183166025983495627459

Offset

1,2

Comments

As n increases, this sequence is approximately geometric with common ratio r = lim_{n->oo} a(n)/a(n-1) = (sqrt(6) + sqrt(7))^4 = 337 + 52*sqrt(42). - Ant King, Jan 03 2012

References

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 44.

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).

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)

Mathematica

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

Prog

(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

Crossrefs

Cf. A048922, A048924.

Sequence in context: A253527 A183964 A221816 * A329292 A252535 A390296

Adjacent sequences: A048920 A048921 A048922 * A048924 A048925 A048926

Keyword

nonn,easy

Author

Eric W. Weisstein

Status

approved