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A222393 Nonnegative integers m such that 18*m*(m+1)+1 is a square.

%I A222393 #23 Sep 08 2022 08:46:04
%S A222393 0,4,12,152,424,5180,14420,175984,489872,5978292,16641244,203085960,
%T A222393 565312440,6898944364,19203981732,234361022432,652370066464,
%U A222393 7961375818340,22161378278060,270452416801144,752834491387592,9187420795420572,25574211328900084
%N A222393 Nonnegative integers m such that 18*m*(m+1)+1 is a square.
%C A222393 a(n+2)/a(n) tends to A156164.
%C A222393 a(n) is congruent to {0,2,4} (mod 5, 6 and 10).
%H A222393 Bruno Berselli, <a href="/A222393/b222393.txt">Table of n, a(n) for n = 1..500</a>
%H A222393 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,34,-34,-1,1).
%F A222393 G.f.: 4*x*(1+x)^2/((1-x)*(1-6*x+x^2)*(1+6*x+x^2)).
%F A222393 a(n) = a(-n+1) = a(n-1)+34*a(n-2)-34*a(n-3)-a(n-4)+a(n-5).
%F A222393 a(n) = -1/2+((3+t*(-1)^n)*(3-2*t)^(2*floor(n/2))+(3-t*(-1)^n)*(3+2*t)^(2*floor(n/2)))/12, where t=sqrt(2).
%t A222393 LinearRecurrence[{1, 34, -34, -1, 1}, {0, 4, 12, 152, 424}, 23]
%t A222393 CoefficientList[Series[4 x (1 + x)^2 / ((1 - x) (1 - 6 x + x^2) (1 + 6 x + x^2)), {x, 0, 25}], x] (* _Vincenzo Librandi_, Aug 18 2013 *)
%o A222393 (Magma) m:=22; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(4*(1+x)^2/((1-x)*(1-6*x+x^2)*(1+6*x+x^2))));
%o A222393 (Magma) I:=[0,4,12,152,424]; [n le 5 select I[n] else Self(n-1)+34*Self(n-2)-34*Self(n-3)-Self(n-4)+Self(n-5): n in [1..25]]; // _Vincenzo Librandi_, Aug 18 2013
%o A222393 (Maxima) makelist(expand(-1/2+((3+sqrt(2)*(-1)^n)*(3-2*sqrt(2))^(2*floor(n/2))+(3-sqrt(2)*(-1)^n)*(3+2*sqrt(2))^(2*floor(n/2)))/12), n, 1, 23);
%o A222393 (PARI) x='x+O('x^30); concat([0], Vec(4*x*(1+x)^2/((1-x)*(1-6*x+x^2)*(1+6*x+x^2)))) \\ _G. C. Greubel_, Jul 15 2018
%Y A222393 Cf. nonnegative integers n such that k*n*(n+1)+1 is a square: A001652 (k=2), A001921 (k=3), A001477 (k=4), A053606 (k=5), A105038 (k=6), A105040 (k=7), A053141 (k=8), A222390 (k=10), A105838 (k=11), A061278 (k=12), A104240 (k=13); A105063 (k=17), this sequence (k=18), A101180 (k=19), A077259 (k=20) [incomplete list].
%K A222393 nonn,easy
%O A222393 1,2
%A A222393 _Bruno Berselli_, Feb 19 2013