A104240 Nonnegative integers n such that 13*n^2 + 13*n + 1 is a square.

0, 7, 144, 504, 9727, 187560, 654840, 12626287, 243453384, 849982464, 16388911447, 316002305520, 1103276584080, 21272794432567, 410170749112224, 1432052156154024, 27612070784561167, 532401316345361880, 1858802595411339720, 35840446605565962847

Offset: 0

Author

Gerald McGarvey, Apr 02 2005

Keywords

• nonn
• easy

Comments

The next terms appear to be 243453384, 849982464, 16388911447 (confirmed by Pierre CAMI).

Links

• Index entries for linear recurrences with constant coefficients, signature (1,0,1298,-1298,0,-1,1).

Crossrefs

Cf. similar sequences indexed in A222390. [Bruno Berselli, Feb 19 2013]

Programs

Magma

m:=19; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!((7+137*x+360*x^2+137*x^3+7*x^4)/((1-x)*(1-11*x+x^2)*(1+11*x+120*x^2+11*x^3+x^4)))); // Bruno Berselli, Feb 19 2013

Mathematica

LinearRecurrence[{1, 0, 1298, -1298, 0, -1, 1}, {0, 7, 144, 504, 9727, 187560, 654840}, 20] (* Bruno Berselli, Feb 19 2013 *)

PARI

for(n=0,12626287,if(issquare(13*n*(n+1)+1),print1(n,",")))

Formula

a(0)=0, a(1)=7, a(2)=144, a(3)=504, a(4)=9727, a(6)=187560 and then a(n) = 1298*a(n-3)+648-a(n-6). - Pierre CAMI, Apr 05 2005

G.f.: x*(7+137*x+360*x^2+137*x^3+7*x^4)/((1-x)*(1-11*x+x^2)*(1+11*x+120*x^2+11*x^3+x^4)). - Bruno Berselli, Feb 19 2013

a(n) = a(n-1)+1298*a(n-3)-1298*a(n-4)-a(n-6)+a(n-7). - Bruno Berselli, Feb 19 2013

Extensions

More terms from Pierre CAMI, Apr 05 2005