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A120862 Fixed-j dispersion for Q = 13: array D(g,h) (g, h >= 1), read by ascending antidiagonals.

%I A120862 #26 Jul 11 2020 03:06:31
%S A120862 1,2,10,3,20,109,4,30,218,1189,5,43,327,2378,12970,6,53,469,3567,
%T A120862 25940,141481,7,63,578,5116,38910,282962,1543321,8,76,687,6305,55807,
%U A120862 424443,3086642,16835050,9,86,829,7494,68777,608761,4629963,33670100,183642229
%N A120862 Fixed-j dispersion for Q = 13: array D(g,h) (g, h >= 1), read by ascending antidiagonals.
%C A120862 For each positive integer n, there exists a unique pair (j,k) of positive integers such that (j+k+1)^2 - 4*k = 13*n^2; in fact, j(n) = A120869(n) and k(n) = A120870(n).
%C A120862 Suppose g >= 1 and let j = j(g). The numbers in row g of array D are among those n for which (j+k+1)^2 - 4*k = 13*n^2 for some k; that is, j stays fixed and k and n vary - hence the name "fixed-j dispersion". (The fixed-k dispersion for Q=13 is A120863.)
%C A120862 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 A120862 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 A120862 N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>.
%F A120862 Define f(n) = floor(r*n) - floor(3*F(n)), where r = (11 + 3*sqrt(13))/2 and F(n) is the fractional part of (1 + sqrt(13))*n/2. Let D(g,h) be the term in row g and column h of the array to be defined:
%F A120862 D(1,1) = 1; D(1,2) = f(1); and D(1,h) = 11*D(1,h-1) - D(1,h-2) for h >= 3.
%F A120862 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) = 11*D(g+1,h-1) - D(g+1,h-2) for h >= 3. All rows after row 1 are thus inductively defined. [Corrected using Conjecture 1 in Kimberling (2007) by _Petros Hadjicostas_, Jul 07 2020]
%e A120862 Northwest corner:
%e A120862   1, 10, 109, 1189, ...
%e A120862   2, 20, 218, 2378, ...
%e A120862   3, 30, 327, 3567, ...
%e A120862   4, 43, 469, 5116, ...
%e A120862   5, 53, 578, 6305, ...
%e A120862   6, 63, 687, 7494, ...
%e A120862   ...
%o A120862 (PARI) f(n) = floor((11 + 3*sqrt(13))/2*n) - floor(3*frac((1 + sqrt(13))*n/2));
%o A120862 unused(listus) = {my(v=vecsort(Vec(listus))); for (i=1, vecmax(v), if (!vecsearch(v, i), return (i)); ); };
%o A120862 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 = 11*m[g, h-1] - m[g, h-2]; m[g, h] = t; listput(listus, t); ); ); m; };
%o A120862 lista(nb) = {my(m=D(nb)); for (n=1, nb, for (j=1, n, print1(m[n-j+1, j], ", ");););} \\ _Michel Marcus_, Jul 09 2020
%Y A120862 Cf. A120858, A120859, A120860, A120861, A120863, A120869, A120870.
%K A120862 nonn,tabl
%O A120862 1,2
%A A120862 _Clark Kimberling_, Jul 09 2006
%E A120862 Name edited by _Petros Hadjicostas_, Jul 07 2020
%E A120862 More terms from _Michel Marcus_, Jul 09 2020