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A120860 Fixed-j dispersion for Q=8. 6
1, 2, 5, 3, 10, 29, 4, 17, 58, 169, 6, 22, 99, 338, 985, 7, 34, 128, 577, 1970, 5741, 8, 39, 198, 746, 3363, 11482, 33461, 9, 46, 227, 1154, 4348, 19601, 66922, 195025, 11, 51, 268, 1323, 6726, 25342, 114243, 390050, 1136689, 12, 63, 297, 1562, 7711 (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), 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.

REFERENCES

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) 1-17.

LINKS

Table of n, a(n) for n=1..50.

N. J. A. Sloane, Classic Sequences.

FORMULA

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.

EXAMPLE

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.

CROSSREFS

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

Sequence in context: A277709 A138765 A097753 * A259971 A091809 A110315

Adjacent sequences:  A120857 A120858 A120859 * A120861 A120862 A120863

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jul 09 2006

STATUS

approved

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Last modified June 2 11:35 EDT 2020. Contains 334771 sequences. (Running on oeis4.)