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)=A0870590(n). Suppose g>=1 and let k=k(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 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 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.

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*sqrt(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 7 41 239

2 12 70 408

3 19 111 667

4 24 140 816

5 31 181 1055.

Cf. A120858, A120859, A120860, A120862, A120863, A087056, A087059.

Sequence in context: A051430 A091578 A056756 this_sequence A099130 A076992 A138751

Adjacent sequences: A120858 A120859 A120860 this_sequence A120862 A120863 A120864

nonn,tabl

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