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

0, 9, 360, 13689, 519840, 19740249, 749609640, 28465426089, 1080936581760, 41047124680809, 1558709801289000, 59189925324301209, 2247658452522156960, 85351831270517663289, 3241121929827149048040, 123077281502161146162249

Offset: 0

Author

Mohamed Bouhamida, Nov 14 2007

Keywords

• nonn
• easy

Comments

Also, numbers n such that 5*A000217(n) is a square. [Bruno Berselli, Dec 16 2013]

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500
• Kenneth M. Wilke, Problem 269, Crux Mathematicorum, Vol. 3, No. 7 (1977), p. 190; Solution to Problem 269 by Lindsay Reynolds, W. J. Blundon and M. S. Klamkin, ibid., Vol. 4, No. 3 (1978), pp. 79-82; Comment by the MaScoT Problems Group, ibid., Vol. 6, No. 2 (1980), pp. 44-46.
• Index entries for linear recurrences with constant coefficients, signature (39,-39,1).

Crossrefs

Cf. A007654, A078986.

Cf. A233474 (numbers n such that 5*A000217(n)-1 is a square).

Programs

Mathematica

LinearRecurrence[{39,-39,1},{0,9,360},30] (* Harvey P. Dale, Jun 01 2014 *)

Formula

a(0)=0, a(1)=9 and a(n) = 38*a(n-1) - a(n-2) + 18.

a(n) = (A078986(n) - 1)/2. - Max Alekseyev, Nov 13 2009

G.f.: -9*x*(x+1)/((x-1)*(x^2-38*x+1)). - Colin Barker, Oct 24 2012

From Amiram Eldar, Feb 15 2022: (Start)

sqrt(a(n)+1) - sqrt(n) = (sqrt(10)-3)^n (Wilke, 1977).

a(n) = ((Sum_{k=0..n} binomial(2*n, 2*k) * 10^(n-k) * 9*k)- 1)/2 (Klamkin, 1978).

a(n) = sinh(n*log(sqrt(10)+3))^2 (MaScoT Problems Group, 1980). (End)

Extensions

More terms from Max Alekseyev, Nov 13 2009