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A120870 a(n) is the number k for which there exists a unique pair (j,k) of positive integers such that (j + k + 1)^2 - 4*k = 13*n^2.

%I A120870 #15 Jul 09 2020 07:23:50
%S A120870 3,3,1,12,9,4,23,17,9,36,27,16,3,39,25,9,53,36,17,69,49,27,3,64,39,12,
%T A120870 81,53,23,100,69,36,1,87,51,13,107,68,27,129,87,43,153,108,61,12,131,
%U A120870 81,29,156,103,48,183,127,69,9,153,92,29,181,117,51,211,144,75,4,173,101
%N A120870 a(n) is the number k for which there exists a unique pair (j,k) of positive integers such that (j + k + 1)^2 - 4*k = 13*n^2.
%C A120870 The j's that match these k's comprise A120869.
%H A120870 Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling2/kimberling45.html">The equation (j+k+1)^2-4*k = Q*n^2 and related dispersions</a>, Journal of Integer Sequences, 10 (2007), Article #07.2.7.
%F A120870 Let r = (1/2)*sqrt(13). If n is odd, then a(n) = ([n*r+1/2] + 1/2)^2 - (13/4)*n^2; if n is even, then a(n) = (1 + [n*r])^2 - (13/4)*n^2, where [ ] is the floor function. [corrected by _Michel Marcus_, Jul 08 2020]
%F A120870 (A120869(n) + a(n) + 1)^2 - 4*a(n) = 13*n^2. - _Petros Hadjicostas_, Jul 08 2020
%e A120870 3 = ([1*r + 1/2] + 1/2)^2 - (13/4)*1^2,
%e A120870 3 = (1 + [2*r])^2 - (13/4)*2^2,
%e A120870 1 = ([3*r+1/2] + 1/2)^2 - (13/4)*3^2, etc.
%e A120870 Moreover,
%e A120870 for n = 1, the unique (j,k) is (1,3): (1 + 3 + 1)^2 - 4*3 = 13*1;
%e A120870 for n = 2, the unique (j,k) is (4,3): (4 + 3 + 1)^2 - 4*3 = 13*4;
%e A120870 for n = 3, the unique (j,k) is (9,1): (9 + 1 + 1)^2 - 4*1 = 13*9.
%o A120870 (PARI) a(n) = my(r = sqrt(13)/2); if (n%2, (floor(n*r+1/2) + 1/2)^2 - (13/4)*n^2, (1 + floor(n*r))^2 - (13/4)*n^2); \\ _Michel Marcus_, Jul 08 2020
%Y A120870 Cf. A120862, A120863, A120869.
%K A120870 nonn
%O A120870 1,1
%A A120870 _Clark Kimberling_, Jul 09 2006