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A120860 Fixed-j dispersion for Q = 8: Square array D(g,h) (g, h >= 1), read by ascending antidiagonals.

%I A120860 #34 Jul 11 2020 23:53:13
%S A120860 1,2,5,3,10,29,4,17,58,169,6,22,99,338,985,7,34,128,577,1970,5741,8,
%T A120860 39,198,746,3363,11482,33461,9,46,227,1154,4348,19601,66922,195025,11,
%U A120860 51,268,1323,6726,25342,114243,390050,1136689,12,63,297,1562,7711
%N A120860 Fixed-j dispersion for Q = 8: Square array D(g,h) (g, h >= 1), read by ascending antidiagonals.
%C A120860 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). Suppose g >= 1 and let j = j(g).
%C A120860 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 k; that is, j stays fixed and k and n vary - hence the name "fixed-j dispersion". (The fixed-k dispersion for Q = 8 is A120861.)
%C A120860 Every positive integer occurs exactly once in 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. Possibly, D is the dispersion of A098021.
%C A120860 From _Petros Hadjicostas_, Jul 08 2020: (Start)
%C A120860 It appears that the first column of the dispersion array D is A187968. That is, the first column of D consists of those positive integers m such that A187967(m) = 1; i.e., those m for which floor(sqrt(2)*m + 2*sqrt(2)) - floor(sqrt(2)*m) - floor(2*sqrt(2)) = 1.
%C A120860 Indeed, _Clark Kimberling_, the author of this sequence and of sequence A187968, conjectured that A187968 is the complement of A098021, and that A098021 is the complement of the first column of array D here. (End)
%H A120860 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 A120860 N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>.
%F A120860 Define f(n) = 3*n + 2*floor(n*2^(1/2)). Let D(g,h) be the term in row g and column h of the array to be defined:
%F A120860 D(1,1) = 1; D(1,2) = f(1); D(1,h) = 6*D(1,h-1) - D(1,h-2) for h >= 3.
%F A120860 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 A120860 Northwest corner:
%e A120860   1,  5,  29,  169,  985,  5741,  33461,  195025, ...
%e A120860   2, 10,  58,  338, 1970, 11482,  66922,  390050, ...
%e A120860   3, 17,  99,  577, 3363, 19601, 114243,  665857, ...
%e A120860   4, 22, 128,  746, 4348, 25342, 147704,  860882, ...
%e A120860   6, 34, 198, 1154, 6726, 39202, 228486, 1331714, ...
%e A120860   7, 39, 227, 1323, 7711, 44943, 261947, 1526739, ...
%e A120860   ...
%o A120860 (PARI) f(n) = 3*n + 2*sqrtint(2*n^2); \\ A098021
%o A120860 unused(listus) = {my(v=vecsort(Vec(listus))); for (i=1, vecmax(v), if (!vecsearch(v, i), return (i)););};
%o A120860 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 A120860 Cf. A087056, A087059, A098021, A120858, A120859, A120861, A120862, A120863, A120871, A187967, A187968.
%K A120860 nonn,tabl
%O A120860 1,2
%A A120860 _Clark Kimberling_, Jul 09 2006
%E A120860 Name edited by _Petros Hadjicostas_, Jul 07 2020