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%I #16 Sep 08 2022 08:45:17
%S 0,7,144,504,9727,187560,654840,12626287,243453384,849982464,
%T 16388911447,316002305520,1103276584080,21272794432567,
%U 410170749112224,1432052156154024,27612070784561167,532401316345361880,1858802595411339720,35840446605565962847
%N Nonnegative integers n such that 13*n^2 + 13*n + 1 is a square.
%C The next terms appear to be 243453384, 849982464, 16388911447 (confirmed by _Pierre CAMI_).
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1298,-1298,0,-1,1).
%F a(0)=0, a(1)=7, a(2)=144, a(3)=504, a(4)=9727, a(6)=187560 and then a(n) = 1298*a(n-3)+648-a(n-6). - _Pierre CAMI_, Apr 05 2005
%F G.f.: x*(7+137*x+360*x^2+137*x^3+7*x^4)/((1-x)*(1-11*x+x^2)*(1+11*x+120*x^2+11*x^3+x^4)). - _Bruno Berselli_, Feb 19 2013
%F a(n) = a(n-1)+1298*a(n-3)-1298*a(n-4)-a(n-6)+a(n-7). - _Bruno Berselli_, Feb 19 2013
%t LinearRecurrence[{1, 0, 1298, -1298, 0, -1, 1}, {0, 7, 144, 504, 9727, 187560, 654840}, 20] (* _Bruno Berselli_, Feb 19 2013 *)
%o (PARI) for(n=0,12626287,if(issquare(13*n*(n+1)+1),print1(n,",")))
%o (Magma)
%o m:=19; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((7+137*x+360*x^2+137*x^3+7*x^4)/((1-x)*(1-11*x+x^2)*(1+11*x+120*x^2+11*x^3+x^4)))); // _Bruno Berselli_, Feb 19 2013
%Y Cf. similar sequences indexed in A222390. [_Bruno Berselli_, Feb 19 2013]
%K nonn,easy
%O 0,2
%A _Gerald McGarvey_, Apr 02 2005
%E More terms from _Pierre CAMI_, Apr 05 2005
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