A048919 Indices of 9-gonal numbers which are also heptagonal.

1, 88, 12445, 1767052, 250908889, 35627295136, 5058825000373, 718317522757780, 101996029406604337, 14482717858215058024, 2056443939837131635021, 292000556739014477114908, 41462022613000218618681865, 5887315210489292029375709872, 835957297866866467952732119909

Offset

1,2

Comments

As n increases, this sequence is approximately geometric with common ratio r = lim_{n->oo} a(n)/a(n-1) = (6 + sqrt(35))^2 = 71 + 12*sqrt(35). - Ant King, Jan 01 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 Heptagonal number.

Index entries for linear recurrences with constant coefficients, signature (143,-143,1).

Formula

G.f.: -x*(1 - 55*x + 4*x^2) / ( (x-1)*(x^2 - 142*x + 1) ). - R. J. Mathar, Dec 21 2011

a(n) = (50 + (25-3r)*(6+r)^(2n-1) + (25+3r)*(6-r)^(2n-1))/140, where r=sqrt(35). - Bruno Berselli, Dec 21 2011

From Ant King, Jan 01 2012: (Start)

a(n) = 142*a(n-1) - a(n-2) - 50.

a(n) = ceiling(1/140*(45 + 7*sqrt(35))*(6 + sqrt(35))^(2*n - 2)). (End)

E.g.f.: exp(x)*(25 + exp(70*x)*(255*cosh(12*sqrt(35)*x) - 43*sqrt(35)*sinh(12*sqrt(35)*x)))/70 - 4. - Stefano Spezia, Oct 11 2025

Mathematica

LinearRecurrence[{143, -143, 1}, {1, 88, 12445}, 30] (* Vincenzo Librandi, Dec 21 2011 *)

Prog

(Magma) I:=[1, 88, 12445]; [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
(Maxima) makelist(expand((50+(25-3*sqrt(35))*(6+sqrt(35))^(2*n-1)+(25+3*sqrt(35))*(6-sqrt(35))^(2*n-1))/140), n, 1, 12); /* Bruno Berselli, Dec 21 2011 */

Crossrefs

Cf. A048920, A048921.

Sequence in context: A052069 A346926 A174499 * A159718 A157460 A267917

Adjacent sequences: A048916 A048917 A048918 * A048920 A048921 A048922

Keyword

nonn,easy

Author

Eric W. Weisstein

Status

approved