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A120869 a(n) is the number j for which there exists a unique pair (j,k) of positive integers such that (j + k + 1)^2 - 4*k = 13*n^2.

%I A120869 #21 Jul 11 2020 03:05:12
%S A120869 1,4,9,3,9,17,3,12,23,1,13,27,43,12,29,48,9,29,51,4,27,52,79,23,51,81,
%T A120869 17,48,81,9,43,79,117,36,75,116,27,69,113,16,61,108,3,51,101,153,39,
%U A120869 92,147,25,81,139,9,68,129,192,53,117,183,36,103,172,17,87,159,233,69,144
%N A120869 a(n) is the number j for which there exists a unique pair (j,k) of positive integers such that (j + k + 1)^2 - 4*k = 13*n^2.
%C A120869 The k's that match these j's comprise A120870.
%H A120869 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 A120869 Let r = (1/2)*sqrt(13). If n is odd, then a(n) = 13*(n/2)^2 - ([n*r+1/2] - 1/2)^2; if n is even, then a(n) = 13*(n/2)^2 - [n*r]^2, where [ ] is the floor function. [corrected by _Michel Marcus_, Jul 08 2020]
%F A120869 (a(n) + A120870(n) + 1)^2 - 4*A120870(n) = 13*n^2. - _Petros Hadjicostas_, Jul 08 2020
%e A120869 1 = 13*(1/2)^2 - ([1*r + 1/2] - 1/2)^2,
%e A120869 4 = 13*(2/2)^2 - ([2*r])^2,
%e A120869 9 = 13*(3/2)^2 - ([3*r + 1/2]- 1/2)^2, etc.
%e A120869 Moreover,
%e A120869 for n = 1, the unique (j,k) is (1,3): (1 + 3 + 1)^2 - 4*3 = 13*1;
%e A120869 for n = 2, the unique (j,k) is (4,3): (4 + 3 + 1)^2 - 4*3 = 13*4;
%e A120869 for n = 3, the unique (j,k) is (9,1): (9 + 1 + 1)^2 - 4*1 = 13*9.
%o A120869 (PARI) a(n) = my(r = sqrt(13)/2); if (n%2, 13*(n/2)^2 - (floor(n*r+1/2) - 1/2)^2, 13*(n/2)^2 - floor(n*r)^2); \\ _Michel Marcus_, Jul 08 2020
%Y A120869 Cf. A120862, A120863, A120870.
%K A120869 nonn
%O A120869 1,2
%A A120869 _Clark Kimberling_, Jul 09 2006