A048918 - OEIS

%I #25 Aug 02 2023 07:17:05

%S 1,325,5330229625,1353857339341,22184715227362706161,

%T 5634830324997758086741,92334031424171069457850940521,

%U 23452480456295952079681300143325,384299427405961840930468013697980089825,97610541547790513644729906482502335077221

%N 9-gonal hexagonal numbers.

%C 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

%H Colin Barker, <a href="/A048918/b048918.txt">Table of n, a(n) for n = 1..208</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NonagonalHexagonalNumber.html">Nonagonal Hexagonal Number.</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,4162056194,-4162056194,-1,1).

%F From _Ant King_, Dec 28 2011: (Start)

%F 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)).

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

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

%F 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).

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

%t LinearRecurrence[{1, 4162056194, -4162056194, -1, 1},  {1, 325, 5330229625, 1353857339341, 22184715227362706161}, 8] (* _Ant King_, Dec 27 2011 *)

%o (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

%Y Cf. A048916, A048917.

%K nonn,easy

%O 1,2

%A _Eric W. Weisstein_