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 A120862 Fixed-j dispersion for Q=13. 5

%I

%S 1,2,10,3,20,109,4,30,218,1189,5,43,327,2378,12970,6,53,469,3567,

%T 25940,141481,7,63,578,5116,38910,282962,1543321,8,76,687,6305,55807

%N Fixed-j dispersion for Q=13.

%C 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.

%D 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.

%H N. J. A. Sloane, <a href="/classic.html#WYTH">Classic Sequences</a>.

%F 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.

%e Northwest corner:

%e 1 10 109 1189

%e 2 20 218 2378

%e 3 30 327 3567

%e 4 43 469 5116

%e 5 53 578 6353.

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

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Jul 09 2006

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