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 A120861 Fixed-k dispersion for Q = 8: Square array D(g,h) (g, h >= 1), read by ascending antidiagonals. 9
 1, 2, 7, 3, 12, 41, 4, 19, 70, 239, 5, 24, 111, 408, 1393, 6, 31, 140, 647, 2378, 8119, 8, 36, 181, 816, 3771, 13860, 47321, 9, 48, 210, 1055, 4756, 21979, 80782, 275807, 10, 53, 280, 1224, 6149, 27720, 128103, 470832, 1607521, 11, 60, 309, 1632, 7134 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 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.) 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. LINKS Clark Kimberling, The equation (j+k+1)^2 - 4*k = Q*n^2 and related dispersions, Journal of Integer Sequences, 10 (2007), Article #07.2.7. N. J. A. Sloane, Classic Sequences. FORMULA 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: 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. 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] EXAMPLE Northwest corner:   1,  7,  41,  239, 1393,  8119,  47321, ...   2, 12,  70,  408, 2378, 13860,  80782, ...   3, 19, 111,  647, 3771, 21979, 128103, ...   4, 24, 140,  816, 4756, 27720, 161564, ...   5, 31, 181, 1055, 6149, 35839, 208885, ...   6, 36, 210, 1224, 7134, 41580, 242346, ... ... [Edited by Petros Hadjicostas, Jul 07 2020] PROG (PARI) f(n) = 3*n + 2*sqrtint(2*n^2) + 2; unused(listus) = {my(v=vecsort(Vec(listus))); for (i=1, vecmax(v), if (!vecsearch(v, i), return (i)); ); }; 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 CROSSREFS Cf. A087056, A087059, A120858, A120859, A120860, A120862, A120863, A336109 (first column), A002315 (first row), A001542 (2nd row), A253811 (3rd row). Sequence in context: A258249 A256448 A056756 * A236542 A279357 A099130 Adjacent sequences:  A120858 A120859 A120860 * A120862 A120863 A120864 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Jul 09 2006 EXTENSIONS Name edited by Petros Hadjicostas, Jul 07 2020 STATUS approved

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Last modified September 27 16:29 EDT 2021. Contains 347691 sequences. (Running on oeis4.)