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.

1, 2, 6, 3, 12, 35, 4, 18, 70, 204, 5, 24, 105, 408, 1189, 7, 30, 140, 612, 2378, 6930, 8, 41, 175, 816, 3567, 13860, 40391, 9, 47, 239, 1020, 4756, 20790, 80782, 235416, 10, 53, 274, 1393, 5945, 27720, 121173, 470832, 1372105, 11, 59, 309, 1597, 8119

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..50.

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.

N. J. A. Sloane, Classic Sequences.

Eric Weisstein's World of Mathematics, Beatty sequence.

Wikipedia, Beatty sequence.

Formula

(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).

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

Example

Northwest corner:
  1,  6,  35,  204, 1189, ...
  2, 12,  70,  408, 2378, ...
  3, 18, 105,  612, 3567, ...
  4, 24, 140,  816, 4756, ...
  5, 30, 175, 1020, 5945, ...
  ... [Corrected by Petros Hadjicostas, Jul 07 2020]
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.

Prog

(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}
/* To print the array flattened */
flat(nn)={D=tabls(nn); for(n=1, nn, for(m=1, n, print1(D[n+1-m, m], ", ")))}
/* To print the square array */
square(nn)={D=tabls(nn); for(n=1, nn, for(m=1, nn, print1(D[n, m], ", ")); print())} // Petros Hadjicostas, Jul 07 2020

Crossrefs

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

Sequence in context: A207901 A054619 A054618 * A253258 A098810 A360006

Adjacent sequences: A120856 A120857 A120858 * A120860 A120861 A120862

Keyword

nonn,tabl

Author

Clark Kimberling, Jul 09 2006

Extensions

Name edited by Petros Hadjicostas, Jul 07 2020

Status

approved