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A120863 Fixed-k dispersion for Q=13. 5
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 (list; table; graph; refs; listen; history; text; internal format)



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 k=k(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 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 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.


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.


Table of n, a(n) for n=1..26.

N. J. A. Sloane, Classic Sequences.


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.


Northwest corner:

1 13 142 1549

2 23 251 2738

3 33 360 3927

4 46 502 5476


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

Sequence in context: A249224 A217010 A091023 * A286459 A093079 A095417

Adjacent sequences:  A120860 A120861 A120862 * A120864 A120865 A120866




Clark Kimberling, Jul 09 2006



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