A101180 Numbers n such that 19*n^2 + 19*n + 1 is a square.

0, 8, 671, 15639, 42159, 981911, 77624048, 1807894920, 4873553880, 113507005568, 8973184757831, 208989037004319, 563373081435879, 13121182828725551, 1037282211558181688, 24158714697817430640, 65124801462951244560, 1516782492521509296728

Offset: 1

Author

Pierre CAMI, Apr 06 2005, Apr 22 2005

Keywords

• nonn
• easy

Comments

Define a(1)=0, a(2)=8, a(3)=671, a(4)=15639, a(5)=42159, a(6)=981911, the first 6 terms found for the sequence then a(7)=57799*(2*a(3)+1)-a(2)-1, a(8)=57799*(2*a(4)+1)-a(1)-1 for n>8 a(n)=57799*(2*a(n-4)+1)-a(n-8)-1 remark:57799 = 38*39*39+1 =2*19*(39^2)+1

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..790
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,115598,-115598,0,0,-1,1).

Crossrefs

Cf. A105839.

Programs

Mathematica

CoefficientList[Series[-x^2(8x^6+663x^5+14968x^4+26520x^3+14968x^2+663x+8)/((x-1)(x^4-340x^2+1)(x^4+340x^2+1)),{x,0,30}],x] (* or *) LinearRecurrence[{1,0,0,115598,-115598,0,0,-1,1},{0,0,8,671,15639,42159,981911,77624048,1807894920},30] (* Harvey P. Dale, Aug 16 2025 *)

Formula

G.f.: -x^2*(8*x^6+663*x^5+14968*x^4+26520*x^3+14968*x^2+663*x+8) / ((x-1)*(x^4-340*x^2+1)*(x^4+340*x^2+1)). - Colin Barker, Mar 05 2013

Extensions

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar

More terms (using recursive formula in comment) from Jon E. Schoenfield, Jul 10 2010

a(18) from Colin Barker, Mar 05 2013