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A132593 Nonnegative integer solutions X to the equation: X(X + 1) - 10*Y^2 = 0.

%I A132593 #33 Mar 08 2024 01:12:06
%S A132593 0,9,360,13689,519840,19740249,749609640,28465426089,1080936581760,
%T A132593 41047124680809,1558709801289000,59189925324301209,
%U A132593 2247658452522156960,85351831270517663289,3241121929827149048040,123077281502161146162249
%N A132593 Nonnegative integer solutions X to the equation: X(X + 1) - 10*Y^2 = 0.
%C A132593 Also, numbers n such that 5*A000217(n) is a square. [_Bruno Berselli_, Dec 16 2013]
%H A132593 Seiichi Manyama, <a href="/A132593/b132593.txt">Table of n, a(n) for n = 0..500</a>
%H A132593 Kenneth M. Wilke, <a href="https://cms.math.ca/publications/crux/issue/?volume=3&amp;issue=7">Problem 269</a>, Crux Mathematicorum, Vol. 3, No. 7 (1977), p. 190; <a href="https://cms.math.ca/publications/crux/issue/?volume=4&amp;issue=3">Solution to Problem 269</a> by Lindsay Reynolds, W. J. Blundon and M. S. Klamkin, ibid., Vol. 4, No. 3 (1978), pp. 79-82; <a href="https://cms.math.ca/publications/crux/issue?volume=6&amp;issue=2">Comment</a> by the MaScoT Problems Group, ibid., Vol. 6, No. 2 (1980), pp. 44-46.
%H A132593 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (39,-39,1).
%F A132593 a(0)=0, a(1)=9 and a(n) = 38*a(n-1) - a(n-2) + 18.
%F A132593 a(n) = (A078986(n) - 1)/2. - _Max Alekseyev_, Nov 13 2009
%F A132593 G.f.: -9*x*(x+1)/((x-1)*(x^2-38*x+1)). - _Colin Barker_, Oct 24 2012
%F A132593 From _Amiram Eldar_, Feb 15 2022: (Start)
%F A132593 sqrt(a(n)+1) - sqrt(n) = (sqrt(10)-3)^n (Wilke, 1977).
%F A132593 a(n) = ((Sum_{k=0..n} binomial(2*n, 2*k) * 10^(n-k) * 9*k)- 1)/2 (Klamkin, 1978).
%F A132593 a(n) = sinh(n*log(sqrt(10)+3))^2 (MaScoT Problems Group, 1980). (End)
%t A132593 LinearRecurrence[{39,-39,1},{0,9,360},30] (* _Harvey P. Dale_, Jun 01 2014 *)
%Y A132593 Cf. A007654, A078986.
%Y A132593 Cf. A233474 (numbers n such that 5*A000217(n)-1 is a square).
%K A132593 nonn,easy
%O A132593 0,2
%A A132593 _Mohamed Bouhamida_, Nov 14 2007
%E A132593 More terms from _Max Alekseyev_, Nov 13 2009