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A120859 Dispersion of the sequence ([r*n] + 1: n >= 1), where r = 3 + 8^(1/2): square array D(n,m) (n, m >= 1), read by ascending antidiagonals. 5

%I

%S 1,2,6,3,12,35,4,18,70,204,5,24,105,408,1189,7,30,140,612,2378,6930,8,

%T 41,175,816,3567,13860,40391,9,47,239,1020,4756,20790,80782,235416,10,

%U 53,274,1393,5945,27720,121173,470832,1372105,11,59,309,1597,8119

%N Dispersion of the sequence ([r*n] + 1: n >= 1), where r = 3 + 8^(1/2): square array D(n,m) (n, m >= 1), read by ascending antidiagonals.

%C Every positive integer occurs exactly once in array D and every pair of rows of D are mutually interspersed. That is, beginning at the first term of any row of array 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>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BeattySequence.html">Beatty sequence</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Beatty_sequence">Beatty sequence</a>.

%F (1) Column 1 is the sequence ([s*(n-1)] + 1: n >= 1), where 1/r + 1/s = 1. The numbers in all the other columns, arranged in increasing order, form the sequence ([r*n] + 1: n >= 1).

%F (2) Every row satisfies these recurrences: x(n+1) = [r*x(n)] + 1 and x(n+2) = 6*x(n+1) - x(n).

%e Northwest corner:

%e 1, 6, 35, 204, 1189, ...

%e 2, 12, 70, 408, 2378, ...

%e 3, 18, 105, 612, 3567, ...

%e 4, 24, 140, 816, 4756, ...

%e 5, 30, 175, 1020, 5945, ...

%e ... [Corrected by _Petros Hadjicostas_, Jul 07 2020]

%e In row 1, we have 6 = [r] + 1, 35 = [6*r], 204 = [35*r] + 1, etc., where r = 3 + 8^(1/2); each new row starts with the least "new" number n, followed by [n*r] + 1, [[n*r + 1]*r + 1], [[[n*r + 1]*r + 1]*r] + 1, and so on.

%o (PARI) tabls(nn)={default("realprecision", 1000); my(D=matrix(nn, nn)); r = 3 + 8^(1/2); s=r/(r-1); for(n=1, nn, D[n, 1]=floor(s*(n-1))+1); for(m=2, nn, for(n=1, nn, D[n, m]=floor(r*D[n, m-1])+1)); D}

%o /* To print the array flattened */

%o flat(nn)={D=tabls(nn); for(n=1, nn, for(m=1, n, print1(D[n+1-m, m], ", ")))}

%o /* To print the square array */

%o square(nn)={D=tabls(nn); for(n=1, nn, for(m=1, nn, print1(D[n, m], ", ")); print())} // _Petros Hadjicostas_, Jul 07 2020

%Y Cf. A120858, A120860, A120861, A120862, A120863.

%K nonn,tabl,changed

%O 1,2

%A _Clark Kimberling_, Jul 09 2006

%E Name edited by _Petros Hadjicostas_, Jul 07 2020

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Last modified July 9 04:51 EDT 2020. Contains 335538 sequences. (Running on oeis4.)