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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
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 (list; table; graph; refs; listen; history; text; internal format)
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

AUTHOR

Clark Kimberling, Jul 09 2006

EXTENSIONS

Name edited by Petros Hadjicostas, Jul 07 2020

STATUS

approved

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Last modified February 26 02:42 EST 2021. Contains 341619 sequences. (Running on oeis4.)