A120863 - OEIS

%I #24 Jul 11 2020 03:06:17

%S 1,2,13,3,23,142,4,33,251,1549,5,46,360,2738,16897,6,56,502,3927,

%T 29867,184318,7,66,611,5476,42837,325799,2010601,8,79,720,6665,59734,

%U 467280,3553922,21932293,9,89,862,7854,72704,651598,5097243,38767343,239244622

%N Fixed-k dispersion for Q = 13: array D(g,h) (g, h >= 1), read by ascending antidiagonals.

%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) and k(n) = A120870(n).

%C Suppose g >= 1 and let k = k(g). The numbers in row g of array 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.)

%C Every positive integer occurs exactly once in the array D and every pair of rows are mutually interspersed. That is, beginning at the first term of any row of D having greater initial term than that of another row, all the following terms individually separate the individual terms of the other row.

%H Clark Kimberling, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL10/Kimberling2/kimberling45.html">The equation (j+k+1)^2 - 4*k = Q*n^2 and related dispersions</a>, Journal of Integer Sequences, 10 (2007), Article #07.2.7.

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

%F Define f(n) = floor(r*n) - floor(3*F(n)) + 3, where r = (11 + 3*sqrt(13))/2 and F(n) is the fractional part of (1 + sqrt(13))*n/2. Let D(g,h) be the term in row g and column h of the array to be defined:

%F D(1,1) = 1; D(1,2) = f(1); and D(1,h) = 11*D(1,h-1) - D(1,h-2) for h >= 3.

%F 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)); and D(g+1,h) = 11*D(g+1,h-1) - D(g+1,h-2) for h >= 3. All rows after row 1 are thus inductively defined. [Corrected using Conjecture 2 in Kimberling (2007) by _Petros Hadjicostas_, Jul 07 2020]

%e Northwest corner:

%e   1, 13, 142, 1549, ...

%e   2, 23, 251, 2738, ...

%e   3, 33, 360, 3927, ...

%e   4, 46, 502, 5476, ...

%e   5, 56, 611, 6665, ...

%e   6, 66, 720, 7854, ...

%e   ...

%o (PARI) f(n) = floor((11 + 3*sqrt(13))/2*n) - floor(3*frac((1 + sqrt(13))*n/2)) + 3;

%o unused(listus) = {my(v=vecsort(Vec(listus))); for (i=1, vecmax(v), if (!vecsearch(v, i), return (i)); ); };

%o D(nb) = {my(m = matrix(nb, nb), t); my(listus = List); for (g=1, nb, if (g==1, t = 1, t = unused(listus)); m[g, 1]=t; listput(listus, t); t = f(t); m[g, 2]=t; listput(listus, t); for (h=3, nb, t = 11*m[g, h-1] - m[g, h-2]; m[g, h] = t; listput(listus, t); ); ); m; };

%o lista(nb) = {my(m=D(nb)); for (n=1, nb, for (j=1, n, print1(m[n-j+1, j], ", ");););} \\ _Michel Marcus_, Jul 09 2020

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

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Jul 09 2006

%E Name edited by _Petros Hadjicostas_, Jul 07 2020

%E More terms from _Michel Marcus_, Jul 09 2020