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A120858 Dispersion of the Beatty sequence ([r*n]: n >= 1), where r = 3 +  8^(1/2): square array D(n,m) (n, m >= 1), read by ascending antidiagonals. 5

%I

%S 1,2,5,3,11,29,4,17,64,169,6,23,99,373,985,7,34,134,577,2174,5741,8,

%T 40,198,781,3363,12671,33461,9,46,233,1154,4552,19601,73852,195025,10,

%U 52,268,1358,6726,26531,114243,430441,1136689,12,58,303,1562

%N Dispersion of the Beatty sequence ([r*n]: 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 are mutually interspersed. That is, beginning at the first term of any row in 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]: n >= 1) where 1/r + 1/s = 1. The numbers in all the other columns, arranged in increasing order, form the sequence ([r*n]: n >= 1).

%F (2) Every row satisfies these recurrences: x(n+1) = [r*x(n)] and x(n+2) = 6*x(n+1) - x(n). (Here [a] is the floor of number a.)

%e Northwest corner:

%e 1, 5, 29, 169, 985, ...

%e 2, 11, 64, 373, 2174, ...

%e 3, 17, 99, 577, 3363, ...

%e 4, 23, 134, 781, 4552, ...

%e 6, 34, 198, 1154, 6726, ...

%e ...

%e In row 1, we have 5 = [r], 29 = [5*r], 169 = [29*r], etc., where r = 3 + 8^(1/2); each new row starts with the least "new" number n, followed by [n*r], [[n*r]*r], [[[n*r]*r]*r], 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)); for(m=2, nn, for(n=1, nn, D[n,m]=floor(r*D[n,m-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. A120859, 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 11 17:06 EDT 2020. Contains 335626 sequences. (Running on oeis4.)