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