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A120862 Fixed-j dispersion for Q=13. 5
1, 2, 10, 3, 20, 109, 4, 30, 218, 1189, 5, 43, 327, 2378, 12970, 6, 53, 469, 3567, 25940, 141481, 7, 63, 578, 5116, 38910, 282962, 1543321, 8, 76, 687, 6305, 55807 (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 = 13*n^2; in fact, j(n)=A120869(n), k(n)=A120870(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 = 13*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=13 is A120863.) 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.

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..33.

N. J. A. Sloane, Classic Sequences.

FORMULA

Define f(n) = 11*n-3*Floor(n*F)+3, where F is the fractional part of (1+sqrt(13))n/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) = 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)); D(g+1,h) = 11*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 10 109 1189

2 20 218 2378

3 30 327 3567

4 43 469 5116

5 53 578 6353.

CROSSREFS

Cf. A120858, A120859, A120860, A120861, A120863, A120869, A120870.

Sequence in context: A176577 A247476 A245062 * A153273 A276486 A234932

Adjacent sequences:  A120859 A120860 A120861 * A120863 A120864 A120865

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Jul 09 2006

STATUS

approved

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Last modified March 28 07:59 EDT 2020. Contains 333079 sequences. (Running on oeis4.)