A048920 Indices of heptagonal numbers (A000566) which are also 9-gonal.

1, 104, 14725, 2090804, 296879401, 42154784096, 5985682462189, 849924754846700, 120683329505769169, 17136182865064375256, 2433217283509635517141, 345499718075503179058724, 49058526749437941790821625, 6965965298702112231117611984, 989118013888950498876910080061

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, Dec 31 2011

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

From Bruno Berselli, Dec 20 2011: (Start)

G.f.: x*(1 - 39*x - 4*x^2)/((1-x)*(1 - 142*x + x^2)).

a(n) = (42 + (-21+5r)*(6+r)^(2n-1) - (21+5r)*(6-r)^(2n-1))/140, where r=sqrt(35). (End)

From Ant King, Dec 31 2011: (Start)

a(n) = 142*a(n-1) - a(n-2) - 42.

a(n) = ceiling(1/140*(49+9*sqrt(35))*(6+sqrt(35))^(2*n-2)). (End)

E.g.f.: 4 + exp(x)*(21 - exp(70*x)*(301*cosh(12*sqrt(35)*x) - 51*sqrt(35)*sinh(12*sqrt(35)*x)))/70. - Stefano Spezia, Oct 11 2025

Mathematica

LinearRecurrence[{143, -143, 1}, {1, 104, 14725}, 30] (* Vincenzo Librandi, Dec 21 2011 *)

Prog

(Maxima) makelist(expand((42+(-21+5*sqrt(35))*(6+sqrt(35))^(2*n-1)-(21+5*sqrt(35))*(6-sqrt(35))^(2*n-1))/140), n, 1, 12); /* Bruno Berselli, Dec 20 2011 */
(Magma) I:=[1, 104, 14725]; [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

Crossrefs

Cf. A000566, A048919, A048921.

Sequence in context: A206013 A187700 A015272 * A091539 A157874 A282188

Adjacent sequences: A048917 A048918 A048919 * A048921 A048922 A048923

Keyword

nonn,easy

Author

Eric W. Weisstein

Status

approved