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%I #21 Sep 08 2022 08:45:17
%S 0,39,159,15720,63480,6256719,25265079,2490158640,10055438160,
%T 991076882199,4002039122799,394446108956760,1592801515436040,
%U 156988560287908479,633931001104421319,62481052548478618080,252302945638044249120,24867301925734202087559
%N Nonnegative integers n such that 11*n^2 + 11*n + 1 is a square.
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,398,-398,-1,1).
%F a(1)=0, a(2)=39, a(3)=398*a(1)+198-a(2), a(4)=398*a(2)+198-a(1), a(n) = 398*a(n-2) + 198 - a(n-4) for n>4.
%F From _Bruno Berselli_, Feb 20 2013: (Start)
%F G.f.: 3*x*(13+40*x+13*x^2)/((1-x)*(1-20*x+x^2)*(1+20*x+x^2)).
%F a(n) = a(-n+1) = -1/2+((11+2*t*(-1)^n)*(10-3*t)^(2*floor(n/2))+(11-2*t*(-1)^n)*(10+3*t)^(2*floor(n/2)))/44, where t=sqrt(11). (End)
%t LinearRecurrence[{1, 398, -398, -1, 1}, {0, 39, 159, 15720, 63480}, 18] (* _Bruno Berselli_, Feb 20 2013 *)
%o (Magma) m:=17; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(3*(13+40*x+13*x^2)/((1-x)*(1-20*x+x^2)*(1+20*x+x^2)))); // _Bruno Berselli_, Feb 20 2013
%o (Maxima) makelist(expand(-1/2+((11+2*sqrt(11)*(-1)^n)*(10-3*sqrt(11))^(2*floor(n/2))+(11-2*sqrt(11)*(-1)^n)*(10+3*sqrt(11))^(2*floor(n/2)))/44), n, 1, 18); /* _Bruno Berselli_, Feb 20 2013 */
%Y Cf. A105837 (square roots of 11*a(n)^2+11*a(n)+1).
%Y Cf. similar sequences indexed in A222390. [_Bruno Berselli_, Feb 20 2013]
%K nonn,easy
%O 1,2
%A _Pierre CAMI_, Apr 22 2005
%E More terms from _Bruno Berselli_, Feb 20 2013
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