A048916 Indices of 9-gonal numbers which are also hexagonal.

1, 10, 39025, 621946, 2517635809, 40124201194, 162422756519761, 2588572715184730, 10478541711598202305, 166999180107303446986, 676012639819623666961969, 10773785102854001863647034, 43612279434844659538786242721, 695059971958523896124021281450, 2813602594783555725665631995917585

Offset

1,2

Comments

As n increases, the ratio of consecutive terms settles into an approximate 2-cycle with the ratio a(n)/a(n-1) bounded above and below by 2024+765*sqrt(7) and 8+3*sqrt(7) respectively. - Ant King, Dec 29 2011

References

Elena Deza and Michel Marie Deza, Figurate numbers, World Scientific Publishing (2012), page 43.

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Eric Weisstein's World of Mathematics, Nonagonal Hexagonal Number

Index entries for linear recurrences with constant coefficients, signature (1,64514,-64514,-1,1).

Formula

G.f.: -x*(1+9*x-25499*x^2+2295*x^3+154*x^4) / ((x-1)*(x^2-254*x+1)*(x^2+254*x+1)). - R. J. Mathar, Dec 21 2011

From Ant King, Dec 29 2011: (Start)

a(n) = 64514*a(n-2)-a(n-4)-23040.

a(n) = 1/28*(3*((3-sqrt(7)*(-1)^n)*(8+3*sqrt(7))^(2*n-2)+(3+sqrt(7)*(-1)^n)*(8-3*sqrt(7))^(2*n-2))+10).

a(n) = ceiling(3/28*(3-sqrt(7)*(-1)^n)*(8+3*sqrt(7))^(2*n-2)). (End)

Mathematica

LinearRecurrence[{1, 64514, -64514, -1, 1}, {1, 10, 39025, 621946, 2517635809}, 210] (* Vincenzo Librandi, Dec 28 2011 *)

Crossrefs

Cf. A048917, A048918.

Sequence in context: A055321 A092300 A375539 * A366313 A260797 A055310

Adjacent sequences: A048913 A048914 A048915 * A048917 A048918 A048919

Keyword

nonn,easy

Author

Eric W. Weisstein

Status

approved