A103715 Define a(1)=0, a(2)=0, a(3)=1, a(4)=3, a(5)=18, a(6)=22, a(7)=119, a(8)=285 such that from i=1 to 8: 420*a(i)^2 + 420*a(i) + 1 = j(i)^2, j(1)=1, j(2)=1, j(3)=29, j(4)=71, j(5)=379, j(6)=461, j(7)=2449, j(8)=5841. Then a(n) = a(n-8) + 4*sqrt(420*a(n-4)^2 + 420*a(n-4) + 1).

0, 0, 1, 3, 18, 22, 119, 285, 1516, 1844, 9797, 23407, 124334, 151226, 803275, 1919129, 10193912, 12398728, 65858793, 157345211, 835776490, 1016544510, 5399617791, 12900388213, 68523478308, 83344251132, 442702800109

Offset

1,4

Comments

By construction, a(n) is integer so 420*(a(n)^2 + 420*a(n) + 1 = j(n)^2.

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

Formula

From R. J. Mathar, Nov 13 2009: (Start)

a(n) = a(n-1) + 82*a(n-4) - 82*a(n-5) - a(n-8) + a(n-9).

G.f.: x^3*(x^2+1)*(x^4+2*x^3+14*x^2+2*x+1)/((1-x)*(x^8-82*x^4+1)). (End)

Mathematica

Rest[CoefficientList[Series[x^3*(x^2+1)*(x^4+2*x^3+14*x^2+2*x+1)/((1-x)*(x^8-82*x^4+1)), {x, 0, 30}], x]] (* G. C. Greubel, Jul 15 2018 *)

Prog

(PARI) x='x+O('x^30); concat([0, 0], Vec(x^3*(x^2+1)*(x^4+2*x^3+14*x^2 +2*x +1)/((1-x)*(x^8-82*x^4+1)))) \\ G. C. Greubel, Jul 15 2018
(Magma) m:=25; R<x>:=PowerSeriesRing(Integers(), m); [0, 0] cat Coefficients(R!(x^3*(x^2+1)*(x^4+2*x^3+14*x^2+2*x+1)/((1-x)*(x^8- 82*x^4 +1)))); // G. C. Greubel, Jul 15 2018

Crossrefs

Cf. A103200, A053141.

Sequence in context: A291167 A361592 A174029 * A131860 A346421 A263578

Adjacent sequences: A103712 A103713 A103714 * A103716 A103717 A103718

Keyword

nonn

Author

Pierre CAMI, Mar 27 2005

Extensions

Extended by R. J. Mathar, Nov 13 2009

Status

approved