login
A120863
Fixed-k dispersion for Q = 13: array D(g,h) (g, h >= 1), read by ascending antidiagonals.
7
1, 2, 13, 3, 23, 142, 4, 33, 251, 1549, 5, 46, 360, 2738, 16897, 6, 56, 502, 3927, 29867, 184318, 7, 66, 611, 5476, 42837, 325799, 2010601, 8, 79, 720, 6665, 59734, 467280, 3553922, 21932293, 9, 89, 862, 7854, 72704, 651598, 5097243, 38767343, 239244622
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 = 13*n^2; in fact, j(n) = A120869(n) and k(n) = A120870(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 = 13*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 = 13 is A120862.)
Every positive integer occurs exactly once in the array D and every pair of rows are mutually interspersed. That is, beginning at the first term of any row of D 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) = floor(r*n) - floor(3*F(n)) + 3, 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:
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.
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 2 in Kimberling (2007) by Petros Hadjicostas, Jul 07 2020]
EXAMPLE
Northwest corner:
1, 13, 142, 1549, ...
2, 23, 251, 2738, ...
3, 33, 360, 3927, ...
4, 46, 502, 5476, ...
5, 56, 611, 6665, ...
6, 66, 720, 7854, ...
...
PROG
(PARI) f(n) = floor((11 + 3*sqrt(13))/2*n) - floor(3*frac((1 + sqrt(13))*n/2)) + 3;
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 = 11*m[g, h-1] - m[g, h-2]; m[g, h] = t; listput(listus, t); ); ); m; };
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
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jul 09 2006
EXTENSIONS
Name edited by Petros Hadjicostas, Jul 07 2020
More terms from Michel Marcus, Jul 09 2020
STATUS
approved