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
Examples
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.
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.
Programs
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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
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.)
Extensions
Name edited by Petros Hadjicostas, Jul 07 2020
Comments