A122770 Numbers k such that A056109(k) is a square.

0, 6, 88, 1230, 17136, 238678, 3324360, 46302366, 644908768, 8982420390, 125108976696, 1742543253358, 24270496570320, 338044408731126, 4708351225665448, 65578872750585150, 913395867282526656, 12721963269204788038, 177194089901584505880

Offset

0,2

Comments

All terms are even. Sequence is infinite. Corresponding squares are s^2 with s = 1, 11, 153, 2131, 29681, 413403, 5757961, 80198051, 1117014753, 15558008491, 216695104121, 3018173449203, 42037733184721, ... (see A122769).

Numbers m such that the distance from (0,0,-1) to (m,m,m) in R^3 is an integer. - James R. Buddenhagen, Jun 15 2013

Also n such that the sum of the pentagonal numbers P(n) and P(n+1) is equal to the sum of two consecutive triangular numbers. - Colin Barker, Dec 07 2014

Links

Colin Barker, Table of n, a(n) for n = 0..874

Index entries for linear recurrences with constant coefficients, signature (15,-15,1).

Formula

a(n) = ((b+1)*(7+4*b)^n - (b-1)*(7-4*b)^n - 2)/6, where b = sqrt(3).

a(n) = 14*a(n-1) - a(n-2) + 4, with a(0)=0, a(1)=6.

a(n) = 2*A011916(n) = (A001353(n+1)^2 - A001075(n)^2)/2. - Richard R. Forberg, Aug 26 2013

a(n) = 15*a(n-1)-15*a(n-2)+a(n-3). - Colin Barker, Dec 07 2014

G.f.: 2*x*(x-3) / ((x-1)*(x^2-14*x+1)). - Colin Barker, Dec 07 2014

Prog

(PARI) concat(0, Vec(2*x*(x-3) / ((x-1)*(x^2-14*x+1)) + O(x^100))) \\ Colin Barker, Dec 07 2014

Crossrefs

Cf. A056109, A122769, A251730.

Sequence in context: A226426 A210005 A178296 * A265267 A218260 A177567

Adjacent sequences: A122767 A122768 A122769 * A122771 A122772 A122773

Keyword

nonn,easy

Author

Zak Seidov, Oct 21 2006

Extensions

More terms from Colin Barker, Dec 07 2014

Status

approved