A048918 9-gonal hexagonal numbers.

1, 325, 5330229625, 1353857339341, 22184715227362706161, 5634830324997758086741, 92334031424171069457850940521, 23452480456295952079681300143325, 384299427405961840930468013697980089825, 97610541547790513644729906482502335077221, 1599475812185636832479963088306383808688155844441

Offset

1,2

Comments

As n increases, the ratio of consecutive terms forms an approximate 2-cycle with the ratio a(n)/a(n-1) bounded above and below by 8193151+3096720*sqrt(7) and 127+48*sqrt(7) respectively. - Ant King, Dec 27 2011

References

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

Links

Colin Barker, Table of n, a(n) for n = 1..208

Eric Weisstein's World of Mathematics, Nonagonal Hexagonal Number.

Index entries for linear recurrences with constant coefficients, signature (1,4162056194,-4162056194,-1,1).

Formula

From Ant King, Dec 28 2011: (Start)

G.f.: x*(1+324*x+1168173106*x^2+20902860*x^3+82621*x^4) / ((1-x)*(1-64514*x+x^2)*(1+64514*x+x^2)).

a(n) = 4162056194*a(n-2)-a(n-4)+1189158912.

a(n) = a(n-1)+4162056194*a(n-2)-4162056194*a(n-3)-a(n-4)+a(n-5).

a(n) = 1/112*(9*((8-3*sqrt(7)*(-1)^n)*(8+3*sqrt(7))^(4*n-4)+(8+3*sqrt(7)*(-1)^n)*(8-3*sqrt(7))^(4*n-4))-32).

a(n) = floor(9/112*(8-3*sqrt(7)*(-1)^n)*(8+3*sqrt(7))^(4*n-4)). (End)

Mathematica

LinearRecurrence[{1, 4162056194, -4162056194, -1, 1},  {1, 325, 5330229625, 1353857339341, 22184715227362706161}, 8] (* Ant King, Dec 27 2011 *)

Prog

(PARI) Vec(x*(1+324*x+1168173106*x^2+20902860*x^3+82621*x^4)/((1-x)*(1-64514*x+x^2)*(1+64514*x+x^2)) + O(x^20)) \\ Colin Barker, Jun 22 2015

Crossrefs

Cf. A048916, A048917.

Sequence in context: A048909 A097739 A203188 * A274307 A358148 A031516

Adjacent sequences: A048915 A048916 A048917 * A048919 A048920 A048921

Keyword

nonn,easy

Author

Eric W. Weisstein

Status

approved