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A120872 a(n) is the value of k for row n of the fixed-k dispersion for Q = 8.

%I A120872 #18 Jul 11 2020 23:26:21
%S A120872 2,1,7,4,14,9,16,7,25,14,23,8,34,17,47,28,41,18,56,31,46,17,63,32,82,
%T A120872 49,68,31,89,50,71,28,94,49,72,23,97,46,124,71,98,41,127,68,97,34,128,
%U A120872 63,161,94,127,56,162,89,124,47,161,82,119,36,158,73,199,112
%N A120872 a(n) is the value of k for row n of the fixed-k dispersion for Q = 8.
%C A120872 This sequence results from A087059 by deleting duplicates.
%H A120872 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.
%e A120872 For each positive integer n, there is a unique pair (j,k) of positive integers such that (j + k + 1)^2 - 4*k = 8*n^2. This representation is used to define the fixed-k dispersion for Q=8, given by A120861, having northwest corner:
%e A120872   1,  7,  41, 239, ...
%e A120872   2, 12,  70, 408, ...
%e A120872   3, 19, 111, 647, ...
%e A120872   4, 24, 140, 816, ...
%e A120872   ...
%e A120872 The pair (j,k) for each n, shown in the position occupied by n in the above array, is shown here:
%e A120872   (1,2), (17,2),  (43,2),  (673,2), ...
%e A120872   (4,1), (32,1), (196,1), (1152,1), ...
%e A120872   (2,7), (46,7), (306,7), (1822,7), ...
%e A120872   (7,4), (63,4), (391,4), (2303,4), ...
%e A120872   ...
%e A120872 The fixed-k for row 1 is a(1) = 2;
%e A120872 the fixed-k for row 2 is a(2) = 1; etc.
%e A120872 (For example, (46 + 7 + 1)^2 - 4*7 = 8*19^2.)
%o A120872 (PARI) f(n) = 3*n + 2*sqrtint(2*n^2) + 2;
%o A120872 unused(listus) = {my(v=vecsort(Vec(listus))); for (i=1, vecmax(v), if (!vecsearch(v, i), return (i)); ); };
%o A120872 D(nb) = {my(m = matrix(nb, nb), t); my(listus = List); for (g=1, nb, if (g==1, t = 1, t = unused(listus)); m[g, 1]=t; listput(listus, t); t = f(t); m[g, 2]=t; listput(listus, t); for (h=3, nb, t = 6*m[g, h-1] - m[g, h-2]; m[g, h] = t; listput(listus, t); ); ); m; }; \\ A120860
%o A120872 q(n) = (1 + sqrtint(2*n^2))^2 - 2*n^2; \\ A087059
%o A120872 lista(nn) = my(m=D(nn)); vector(nn, n, q(m[n, 1])); \\ _Michel Marcus_, Jul 10 2020
%Y A120872 Cf. A087059, A120861.
%K A120872 nonn
%O A120872 1,1
%A A120872 _Clark Kimberling_, Jul 10 2006
%E A120872 More terms from _Michel Marcus_, Jul 10 2020