A065929 (x,y) = (a(n),a(n+1)) are the solutions of (t(x)+t(y))/(1+xy) = t(3) = 6, where t(n) denotes the n-th triangular number t(n) = n(n+1)/2.

3, 35, 416, 4956, 59055, 703703, 8385380, 99920856, 1190664891, 14188057835, 169066029128, 2014604291700, 24006185471271, 286059621363551, 3408709270891340, 40618451629332528, 484012710281098995, 5767534071743855411, 68726396150645165936

Offset

0,1

Links

Colin Barker, Table of n, a(n) for n = 0..900

J.-P. Ehrmann et al., Problem POLYA002, Integer pairs (x,y) for which (x^2+y^2)/(1+pxy) is an integer.

Index entries for linear recurrences with constant coefficients, signature (13,-13,1).

Formula

a(n) = 2*t(m)*a(n-1)-a(n-2)-1, a(0) = m, a(1) = m^3+m^2-1 with m = 3.

G.f.: (4*x-3) / ((1-12*x+x^2)(x-1)).

a(0)=3, a(1)=35, a(2)=416, a(n) = 13*a(n-1)-13*a(n-2)+a(n-3). - Harvey P. Dale, Mar 23 2012

a(n) = (2+(29-5*sqrt(35))*(6-sqrt(35))^n+(6+sqrt(35))^n*(29+5*sqrt(35)))/20. - Colin Barker, Mar 05 2016

Maple

g := (4*x-3)/(1-12*x+x^2)/(x-1): s := series(g, x, 40): for i from 0 to 30 do printf(`%d, `, coeff(s, x, i)) od: # James A. Sellers, Feb 11 2002

Mathematica

CoefficientList[Series[(4x-3)/((1-12x+x^2)(x-1)), {x, 0, 30}], x] (* or *) LinearRecurrence[{13, -13, 1}, {3, 35, 416}, 30] (* Harvey P. Dale, Mar 23 2012 *)

Prog

(PARI) Vec((4*x-3)/((1-12*x+x^2)*(x-1)) + O(x^25)) \\ Colin Barker, Mar 05 2016

Crossrefs

Cf. A000217 (triangular numbers).

Sequence in context: A221687 A221223 A006767 * A161495 A179135 A100033

Adjacent sequences: A065926 A065927 A065928 * A065930 A065931 A065932

Keyword

easy,nonn

Author

Floor van Lamoen, Nov 29 2001

Extensions

More terms from James A. Sellers, Feb 11 2002

Status

approved