A122769 Numbers k such that k^2 is of the form 3*m^2 + 2*m + 1 (A056109).

1, 11, 153, 2131, 29681, 413403, 5757961, 80198051, 1117014753, 15558008491, 216695104121, 3018173449203, 42037733184721, 585510091136891, 8155103542731753, 113585939507107651, 1582048049556775361

Offset: 1

Author

Zak Seidov, Oct 21 2006

Keywords

• nonn
• easy

Comments

All terms are odd. Sequence is infinite. Corresponding m's are 0, 6, 88, 1230, 17136, 238678, 3324360, 46302366, 644908768, 8982420390, 125108976696, 1742543253358, 24270496570320. s^2 are squares in A056109.

The Diophantine equation A000290(x) = A000326(y) + A000326(y-1) has the solutions x = a(n) and y = (4^n + (1 + sqrt(3))^(4*n - 3) + (1 - sqrt(3))^(4*n - 3))/(3*2^(2*n - 1)). - Bruno Berselli, Mar 04 2013

Links

• Tanya Khovanova, Recursive Sequences
• Valcho Milchev and Tsvetelina Karamfilova, Domino tiling in grid - new dependence, arXiv:1707.09741 [math.HO], 2017.
• Index entries for linear recurrences with constant coefficients, signature (14,-1).

Crossrefs

Cf. A056109.

Programs

Mathematica

LinearRecurrence[{14, -1}, {1, 11}, 17] (* Jean-François Alcover, Jan 07 2019 *)

Formula

Alternatively, with a different offset:

a(0) = 1, a(1) = 11, a(n) = 14*a(n-1) - a(n-2), and

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

G.f.: x*(1 - 3*x)/(1 - 14*x + x^2). - Philippe Deléham, Nov 17 2008

E.g.f.: (1/3)*((9*cosh(4*sqrt(3)*x) - 5*sqrt(3)*sinh(4*sqrt(3)*x))*exp(7*x) - 9). - Franck Maminirina Ramaharo, Jan 07 2019

Extensions

Edited by N. J. A. Sloane, Oct 28 2006