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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
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 (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 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 A081469

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

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Last modified February 25 02:40 EST 2021. Contains 341595 sequences. (Running on oeis4.)