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.

1, 2, 5, 3, 11, 29, 4, 17, 64, 169, 6, 23, 99, 373, 985, 7, 34, 134, 577, 2174, 5741, 8, 40, 198, 781, 3363, 12671, 33461, 9, 46, 233, 1154, 4552, 19601, 73852, 195025, 10, 52, 268, 1358, 6726, 26531, 114243, 430441, 1136689, 12, 58, 303, 1562

Offset

1,2

Comments

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.

Links

Table of n, a(n) for n=1..49.

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]: 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).

(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.)

Example

Northwest corner:
  1,  5,  29,  169,  985, ...
  2, 11,  64,  373, 2174, ...
  3, 17,  99,  577, 3363, ...
  4, 23, 134,  781, 4552, ...
  6, 34, 198, 1154, 6726, ...
  ...
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.

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)); for(m=2, nn, for(n=1, nn, D[n, m]=floor(r*D[n, m-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. A120859, A120860, A120861, A120862, A120863.

Sequence in context: A122442 A225258 A162613 * A124937 A279342 A169852

Adjacent sequences: A120855 A120856 A120857 * A120859 A120860 A120861

Keyword

nonn,tabl,changed

Author

Clark Kimberling, Jul 09 2006

Extensions

Name edited by Petros Hadjicostas, Jul 07 2020

Status

approved