A065928 (x,y) = (a(n),a(n+1)) are the solutions of (t(x)+t(y))/(1+xy) = t(2) = 3, where t(n) denotes the n-th triangular number t(n) = n(n+1)/2.

2, 11, 63, 366, 2132, 12425, 72417, 422076, 2460038, 14338151, 83568867, 487075050, 2838881432, 16546213541, 96438399813, 562084185336, 3276066712202, 19094316087875, 111289829815047, 648644662802406, 3780578146999388, 22034824219193921, 128428367168164137

Offset

0,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

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 (7,-7,1).

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 = 2.

G.f.: (3*x-2)/((1-6*x+x^2)*(x-1)).

a(0)=2, a(1)=11, a(2)=63, a(n)=7*a(n-1)-7*a(n-2)+a(n-3). - Harvey P. Dale, Nov 06 2011

a(n) = (4+(14-11*sqrt(2))*(3-2*sqrt(2))^n+(3+2*sqrt(2))^n*(14+11*sqrt(2)))/16. - Colin Barker, Mar 05 2016

Mathematica

CoefficientList[Series[(3x-2)/((1-6x+x^2)(x-1)), {x, 0, 20}], x] (* or *) LinearRecurrence[{7, -7, 1}, {2, 11, 63}, 20] (* Harvey P. Dale, Nov 06 2011 *)

Prog

(Magma) I:=[2, 11, 63]; [n le 3 select I[n] else 7*Self(n-1)-7*Self(n-2)+Self(n-3): n in [1..20]]; // Vincenzo Librandi, Nov 13 2011
(PARI) Vec((3*x-2)/((1-6*x+x^2)*(x-1)) + O(x^40)) \\ Colin Barker, Mar 05 2016

Crossrefs

Sequence in context: A365131 A002629 A235937 * A188648 A114175 A080049

Adjacent sequences: A065925 A065926 A065927 * A065929 A065930 A065931

Keyword

nonn,easy

Author

Floor van Lamoen, Nov 29 2001

Status

approved