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A120861 Fixed-k dispersion for Q = 8: Square array D(g,h) (g, h >= 1), read by ascending antidiagonals.

%I A120861 #33 Jul 22 2020 09:43:17
%S A120861 1,2,7,3,12,41,4,19,70,239,5,24,111,408,1393,6,31,140,647,2378,8119,8,
%T A120861 36,181,816,3771,13860,47321,9,48,210,1055,4756,21979,80782,275807,10,
%U A120861 53,280,1224,6149,27720,128103,470832,1607521,11,60,309,1632,7134
%N A120861 Fixed-k dispersion for Q = 8: Square array D(g,h) (g, h >= 1), read by ascending antidiagonals.
%C A120861 For each positive integer n, there exists a unique pair (j,k) of positive integers such that (j + k + 1)^2 - 4*k = 8*n^2; in fact, j(n) = A087056(n) and k(n) = A087059(n).
%C A120861 Suppose g >= 1 and let k = k(g). The numbers in row g of array D are among those n for which (j + k + 1)^2 - 4*k = 8*n^2 for some j; that is, k stays fixed and j and n vary - hence the name "fixed-k dispersion". (The fixed-j dispersion for Q = 8 is A120860.)
%C A120861 Every positive integer occurs exactly once in array D and every pair of rows are mutually interspersed. That is, beginning at the first term of any row having greater initial term than that of another row, all the following terms individually separate the individual terms of the other row.
%H A120861 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.
%H A120861 N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>.
%F A120861 Define f(n) = 3*n + 2*floor(n*sqrt(2)) + 2. Let D(g,h) be the term in row g and column h of the array to be defined:
%F A120861 D(1,1) = 1; D(1,2) = f(1); and D(1,h) = 6*D(1,h-1) - D(1,h-2) for h >= 3.
%F A120861 For arbitrary g >= 1, once row g is defined, define D(g+1,1) = least positive integer not in rows 1,2,...,g; D(g+1,2) = f(D(g+1,1)); and D(g+1,h) = 6*D(g+1,h-1) - D(g+1,h-2) for h >= 3. All rows after row 1 are thus inductively defined. [Corrected by _Petros Hadjicostas_, Jul 07 2020]
%e A120861 Northwest corner:
%e A120861   1,  7,  41,  239, 1393,  8119,  47321, ...
%e A120861   2, 12,  70,  408, 2378, 13860,  80782, ...
%e A120861   3, 19, 111,  647, 3771, 21979, 128103, ...
%e A120861   4, 24, 140,  816, 4756, 27720, 161564, ...
%e A120861   5, 31, 181, 1055, 6149, 35839, 208885, ...
%e A120861   6, 36, 210, 1224, 7134, 41580, 242346, ...
%e A120861 ... [Edited by _Petros Hadjicostas_, Jul 07 2020]
%o A120861 (PARI) f(n) = 3*n + 2*sqrtint(2*n^2) + 2;
%o A120861 unused(listus) = {my(v=vecsort(Vec(listus))); for (i=1, vecmax(v), if (!vecsearch(v, i), return (i)););};
%o A120861 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;}; \\ _Michel Marcus_, Jul 08 2020
%Y A120861 Cf. A087056, A087059, A120858, A120859, A120860, A120862, A120863, A336109 (first column), A002315 (first row), A001542 (2nd row), A253811 (3rd row).
%K A120861 nonn,tabl
%O A120861 1,2
%A A120861 _Clark Kimberling_, Jul 09 2006
%E A120861 Name edited by _Petros Hadjicostas_, Jul 07 2020