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%I #35 Feb 16 2023 05:07:50
%S 0,13,37,76,124,189,261,352,448,565,685,828,972,1141,1309,1504,1696,
%T 1917,2133,2380,2620,2893,3157,3456,3744,4069,4381,4732,5068,5445,
%U 5805,6208,6592,7021,7429,7884,8316,8797,9253,9760,10240,10773,11277,11836,12364,12949,13501,14112,14688
%N Numbers k such that 25*k+36 is a square.
%C Also, numbers of the form 25m^2+12*m, where m = 0,-1,1,-2,2,-3,3,... - _Bruno Berselli_, Apr 07 2013
%H Vincenzo Librandi, <a href="/A222964/b222964.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F G.f.: x^2*(13+24*x+13*x^2)/((1+x)^2*(1-x)^3).
%F a(n) = (50*n*(n-1)+(2*n-1)*(-1)^n+1)/8.
%F a(n) = a(n-1)+2*a(n-2)-2*a(n-3)-a(n-4)+a(n-5).
%F Sum_{n>=2} 1/a(n) = 25/144 - tan(Pi/50)*Pi/12. - _Amiram Eldar_, Feb 16 2023
%t Select[Range[0, 10000], IntegerQ[Sqrt[25 # + 36]]&] (* or *) CoefficientList[Series[x (13 + 24 x + 13 x^2)/((1+x)^2(1-x)^3), {x, 0, 40}], x]
%o (Magma) [n: n in [0..15000] | IsSquare(25*n+36)]; /* or */ I:=[0, 13, 37, 76, 124]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]];
%o (Magma) [0] cat [25*m^2+12*m where m is n*t: t in [-1, 1], n in [1..20]]; // _Bruno Berselli_, Apr 07 2013
%Y Cf. numbers n such that k^2*n+(k+1)^2 is a square: A028552 (k=2), A218864 (k=3), A165717 (k=4).
%Y Cf. numbers of the form k^2*m^2+floor(k^2/2)*m, where m=0,-1,1,-2,2,-3,3,...: A002378 (k=2), A185039 (k=3), A033996 (k=4), this sequence (k=5), A163758 (k=6).
%K nonn,easy
%O 1,2
%A _Vincenzo Librandi_, Apr 07 2013
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