A048917 Indices of hexagonal numbers which are also 9-gonal.

1, 13, 51625, 822757, 3330519121, 53079328957, 214865110504441, 3424359827493013, 13861807735752971425, 220919149857804895597, 894280664049502087991881, 14252378030502065207035717, 57693622746627769968955223281, 919477916038891084908897334813, 3722046376981663287727675186742425

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 - 12*x + 12902*x^2 + 3036*x^3 + 203*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) - 16128.

a(n) = (1/56)*sqrt(7)*(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)) + 2*sqrt(7)).

a(n) = ceiling((3/56)*sqrt(7)*(3 - sqrt(7)*(-1)^n)*(8 + 3*sqrt(7))^(2*n-2)).

(End)

Mathematica

LinearRecurrence[{1, 64514, -64514, -1, 1}, {1, 13, 51625, 822757, 3330519121}, 210] (* Vincenzo Librandi, Dec 27 2011 *)

Crossrefs

Cf. A048916, A048918.

Sequence in context: A220981 A258670 A375538 * A081317 A203675 A189251

Adjacent sequences: A048914 A048915 A048916 * A048918 A048919 A048920

Keyword

nonn,easy

Author

Eric W. Weisstein

Status

approved