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%I #10 Aug 01 2015 10:03:21
%S 1,589,528601,474682789,426264615601,382785150126589,
%T 343740638549061001,308678710631906651989,277193138406813624424801,
%U 248919129610608002826818989,223529101197187579724859027001,200728883955944835984920579427589
%N Indices of 9-gonal (nonagonal) numbers which are also 10-gonal (decagonal).
%C As n increases, this sequence is approximately geometric with common ratio r = lim(n->Infinity, a(n)/a(n-1)) = (2*sqrt(2)+sqrt(7))^4 = 449+120*sqrt(14).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (899, -899, 1).
%F G.f.: x*(1-310*x-11*x^2) / ((1-x)*(1-898*x+x^2)).
%F a(n) = 898*a(n-1)-a(n-2)-320.
%F a(n) = 899*a(n-1)-899*a(n-2)+a(n-3).
%F a(n) = 1/56*((sqrt(2)+2*sqrt(7))*(2*sqrt(2)+sqrt(7))^(4*n-3)+(sqrt(2)-2*sqrt(7))*(2*sqrt(2)-sqrt(7))^(4*n-3)+20).
%F a(n) = ceiling(1/56*(sqrt(2)+2*sqrt(7))*(2*sqrt(2)+sqrt(7))^(4*n-3)).
%e The second number that is both 9-gonal (nonagonal) and 10-gonal (decagonal) is A001106(589) = 1212751. Hence a(2) = 589.
%t LinearRecurrence[{899, -899, 1}, {1, 589, 528601}, 12]
%Y Cf. A203627, A203629, A001107, A001106.
%K nonn,easy
%O 1,2
%A _Ant King_, Jan 06 2012