cp's OEIS Frontend

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

%I A122769 #42 Jan 01 2024 11:50:43
%S A122769 1,11,153,2131,29681,413403,5757961,80198051,1117014753,15558008491,
%T A122769 216695104121,3018173449203,42037733184721,585510091136891,
%U A122769 8155103542731753,113585939507107651,1582048049556775361
%N A122769 Numbers k such that k^2 is of the form 3*m^2 + 2*m + 1 (A056109).
%C A122769 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.
%C A122769 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
%H A122769 Tanya Khovanova, <a href="http://www.tanyakhovanova.com/RecursiveSequences/RecursiveSequences.html">Recursive Sequences</a>
%H A122769 Valcho Milchev and Tsvetelina Karamfilova, <a href="https://arxiv.org/abs/1707.09741">Domino tiling in grid - new dependence</a>, arXiv:1707.09741 [math.HO], 2017.
%H A122769 <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (14,-1).
%F A122769 Alternatively, with a different offset:
%F A122769 a(0) = 1, a(1) = 11, a(n) = 14*a(n-1) - a(n-2), and
%F A122769 a(n) = ((3 - b)*(7 - 4*b)^n + (3 + b)*(7 + 4*b)^n)/6, b = sqrt(3).
%F A122769 G.f.: x*(1 - 3*x)/(1 - 14*x + x^2). - _Philippe Deléham_, Nov 17 2008
%F A122769 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
%t A122769 LinearRecurrence[{14, -1}, {1, 11}, 17] (* _Jean-François Alcover_, Jan 07 2019 *)
%Y A122769 Cf. A056109.
%K A122769 nonn,easy
%O A122769 1,2
%A A122769 _Zak Seidov_, Oct 21 2006
%E A122769 Edited by _N. J. A. Sloane_, Oct 28 2006