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
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
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, Jul 09 2006
EXTENSIONS
Name edited by Petros Hadjicostas, Jul 07 2020
STATUS
approved