|
%I #18 Aug 16 2015 12:03:56
%S 1,14,7189,165026,86968201,1996480214,1052141284189,24153417459626,
%T 12728805169146001,292208042430070814,153993083884187031589,
%U 3535132873165579243826,1863008316102089539013401,42768037207349135261731814,22538674454209995358797089389
%N Indices of 9-gonal numbers which are also pentagonal.
%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="/A048913/b048913.txt">Table of n, a(n) for n = 1..490</a>
%H Eric Weisstein's World of Mathematics, <a href="http://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)-4320.
%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))*(sqrt(21)-3*(-1)^n)*(2*sqrt(7)+3*sqrt(3))^(2n-2)-(2-sqrt(21))*( sqrt(21)+3*(-1)^n)*(2*sqrt(7)-3*sqrt(3))^(2n-2)+30).
%F a(n) = ceiling(1/84*(2+sqrt(21))*(sqrt(21)-3*(-1)^n)*(2*sqrt(7)+3*sqrt(3))^(2n-2)).
%F G.f.: x*(1+13*x-4923*x^2+563*x^3+26*x^4) / ((1-x)*(1-110*x+x^2)*(1+110*x+x^2)).
%F (End)
%t LinearRecurrence[{1, 12098, -12098, -1, 1}, {1, 14, 7189, 165026, 86968201}, 13] (* _Ant King_, Dec 20 2011 *)
%o (PARI) Vec(-x*(26*x^4+563*x^3-4923*x^2+13*x+1)/((x-1)*(x^2-110*x+1)*(x^2+110*x+1)) + O(x^20)) \\ _Colin Barker_, Jun 22 2015
%Y Cf. A048914, A048915.
%K nonn,easy
%O 1,2
%A _Eric W. Weisstein_
| 