

A122769


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


1



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

1,2


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(y1) has the solutions x = a(n) and y = (4^n+(1+sqrt(3))^(4*n3)+(1sqrt(3))^(4*n3))/(3*2^(2*n1)).  Bruno Berselli, Mar 04 2013


LINKS

Table of n, a(n) for n=1..17.
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).


FORMULA

Alternatively, with a different offset:
a(0) = 1, a(1) = 11, a(n) = 14*a(n1)  a(n2), and
a(n) = ((3  b)*(7  4*b)^n + (3 + b)*(7 + 4*b)^n)/6, b=sqrt(3).
a(n) = (1/6)*sqrt(3)*(74*sqrt(3))^n+(1/6)*sqrt(3)*(7+4*sqrt(3))^n+(1/2)*(7+4*sqrt(3))^n+(1/2)*(74*sqrt(3))^n. [Paolo P. Lava, Aug 06 2008]
G.f.: x*(13*x)/(114*x+x^2). [Philippe Deléham, Nov 17 2008]


CROSSREFS

Cf. A056109.
Sequence in context: A176365 A077577 A157186 * A051608 A191369 A223713
Adjacent sequences: A122766 A122767 A122768 * A122770 A122771 A122772


KEYWORD

nonn,easy


AUTHOR

Zak Seidov, Oct 21 2006


EXTENSIONS

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


STATUS

approved



