A120863 Fixed-k dispersion for Q = 13: array D(g,h) (g, h >= 1), read by ascending antidiagonals.

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

Author

Clark Kimberling, Jul 09 2006

Keywords

• nonn
• tabl

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.

Examples

			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, ...
  ...
		

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.

Crossrefs

Cf. A120858, A120859, A120860, A120861, A120862, A120869, A120870.

Programs

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

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]

Extensions

Name edited by Petros Hadjicostas, Jul 07 2020

More terms from Michel Marcus, Jul 09 2020