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%I #18 Sep 08 2022 08:45:26
%S 1,4,9,1,6,13,22,4,13,24,37,9,22,37,1,16,33,52,6,25,46,69,13,36,61,88,
%T 22,49,78,4,33,64,97,13,46,81,118,24,61,100,141,37,78,121,9,52,97,144,
%U 22,69,118,169,37,88,141,1,54,109,166,16,73,132,193,33,94,157,222,52
%N 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 = 12*n^2.
%C The j's that match these k's comprise A120864.
%H Vincenzo Librandi, <a href="/A120865/b120865.txt">Table of n, a(n) for n = 1..10000</a>
%H 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 a(n) = -3*n^2 + floor(1 + n*sqrt(3))^2.
%e 1 = -3*1 + floor(1 + sqrt(3))^2,
%e 4 = -3*4 + floor(1 + 2*sqrt(3))^2,
%e 9 = -3*9 + floor(1 + 3*sqrt(3))^2, etc.
%e Moreover,
%e for n = 1, the unique (j,k) is (2,1): (2+1+1)^2 - 4*1 = 12*1;
%e for n = 2, the unique (j,k) is (3,4): (3+4+1)^2 - 4*4 = 12*4;
%e for n = 3, the unique (j,k) is (2,9): (2+9+1)^2 - 4*9 = 12*9.
%o (Magma) [-3*n^2+Floor(1+n*Sqrt(3))^2: n in [1..70]]; // _Vincenzo Librandi_, Sep 13 2011
%Y Cf. A120864.
%K nonn
%O 1,2
%A _Clark Kimberling_, Jul 09 2006
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