1,2

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), k(n)=A087059(n). Suppose g>=1 and let j=j(g). The numbers in row g of 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.) 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. Kimberling, The equation (j+k+1)^2-4k=Q*n^2 and related dispersions, preprint.

N. J. A. Sloane, Classic Sequences.

Define f(n)=3*n+2*Floor(n*2^(1/2)). Let D(g,h) be the term in row g, column h of the array to be defined: D(1,1)=1; D(1,2)=f(1); 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)); D(g+1,h)=6*D(g+1,h-1)-D(g+1,h) for h>=3. All rows after row 1 are thus inductively defined.

Northwest corner:

1 5 29 169 985

2 10 58 338 1970

3 17 99 577 3363

4 22 128 746 4348

6 34 198 1154 6726.

Cf. A120858, A120859, A120861, A120862, A120863, A098021, A087056, A087059, A120871.

Sequence in context: A101492 A138765 A097753 this_sequence A091809 A110315 A094744

Adjacent sequences: A120857 A120858 A120859 this_sequence A120861 A120862 A120863

nonn,tabl

Clark Kimberling (ck6(AT)evansville.edu), Jul 09 2006