A048914 - OEIS

%I #18 Feb 16 2025 08:32:40

%S 1,21,10981,252081,132846121,3049673901,1607172358861,36894954600201,

%T 19443571064652241,446355157703555781,235228321132990450741,

%U 5400004661002663236321,2845792209623347408410361,65329255942455062129453661,34428393916794935813958094621

%N Indices of pentagonal numbers which are also 9-gonal.

%C From _Ant King_, Dec 20 2011: (Start)

%C lim(n->Infinity, a(2n+1)/a(2n))=1/2*(527+115*sqrt(21))

%C lim(n->Infinity, a(2n)/a(2n-1))=1/2*(23+5*sqrt(21))

%C (End)

%H Colin Barker, <a href="/A048914/b048914.txt">Table of n, a(n) for n = 1..490</a>

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

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,12098,-12098,-1,1).

%F From _Ant King_, Dec 20 2011: (Start)

%F a(n) = 12098*a(n-2)-a(n-4)-2016.

%F a(n) = a(n-1)+12098*a(n-2)-12098*a(n-3)-a(n-4)+a(n-5).

%F a(n) = 1/84*((2+sqrt(21))*(7-sqrt(21)*(-1)^n)*(2*sqrt(7)+3*sqrt(3))^(2n-2)+ (2-sqrt(21))*(7+sqrt(21)*(-1)^n)*(2*sqrt(7)-3*sqrt(3))^(2n-2)+14).

%F a(n) = ceiling(1/84*(2+sqrt(21))*(7-sqrt(21)*(-1)^n)*(2*sqrt(7)+3*sqrt(3))^(2n-2)).

%F G.f.: x*(1+20*x-1138*x^2-860*x^3-39*x^4) / ((1-x)*(1-110*x+x^2)*(1+110*x+x^2))

%F (End)

%t LinearRecurrence[{1, 12098, -12098, -1, 1}, {1, 21, 10981, 252081, 132846121}, 13] (* _Ant King_, Dec 20 2011 *)

%o (PARI) Vec(x*(39*x^4+860*x^3+1138*x^2-20*x-1)/((x-1)*(x^2-110*x+1)*(x^2+110*x+1)) + O(x^20)) \\ _Colin Barker_, Jun 22 2015

%Y Cf. A048913, A048915.

%K nonn,easy,changed

%O 1,2

%A _Eric W. Weisstein_