OFFSET
0,3
COMMENTS
A permutation of the nonnegative integers. - Alzhekeyev Ascar M, Jun 05 2016
The values in array row n, when expressed in binary, have n trailing 1-bits. - Ruud H.G. van Tol, Mar 18 2025
LINKS
FORMULA
From Philippe Deléham, Feb 19 2014: (Start)
A(0,k) = 2*k = A005843(k),
A(1,k) = 4*k + 1 = A016813(k),
A(2,k) = 8*k + 3 = A017101(k),
A(n,0) = A000225(n),
A(n,1) = A153893(n),
A(n,2) = A153894(n),
A(n,3) = A086224(n),
A(n,4) = A052996(n+2),
A(n,5) = A086225(n),
A(n,6) = A198274(n),
A(n,7) = A196305(n),
A(n,8) = A198275(n),
A(n,9) = A198276(n),
A(n,10) = A171389(n). (End)
From Wolfdieter Lang, Jan 31 2019: (Start)
Array A(n, k) = 2^n*(2*k+1) - 1, for n >= 0 and m >= 0.
The triangle is T(n, k) = A(n-k, k) = 2^(n-k)*(2*k+1) - 1, n >= 0, k=0..n.
See also A054582 after subtracting 1. (End)
From Ruud H.G. van Tol, Mar 17 2025: (Start)
A(0, k) is even. For n > 0, A(n, k) is odd and (3 * A(n, k) + 1) / 2 = A(n-1, 3*k+1).
A(n, k) = 2^n - 1 (mod 2^(n+1)) (equivalent to the comment about trailing 1-bits). (End)
EXAMPLE
The array A begins:
0 2 4 6 8 10 12 14 16 18 ...
1 5 9 13 17 21 25 29 33 37 ...
3 11 19 27 35 43 51 59 67 75 ...
7 23 39 55 71 87 103 119 135 151 ...
15 47 79 111 143 175 207 239 271 303 ...
31 95 159 223 287 351 415 479 543 607 ...
... - Philippe Deléham, Feb 19 2014
From Wolfdieter Lang, Jan 31 2019: (Start)
The triangle T begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 ...
0: 0
1: 1 2
2: 3 5 4
3: 7 11 9 6
4: 15 23 19 13 8
5 31 47 39 27 17 10
6: 63 95 79 55 35 21 12
7: 127 191 159 111 71 43 25 14
8: 255 383 319 223 143 87 51 29 16
9: 511 767 639 447 287 175 103 59 33 18
10: 1023 1535 1279 895 575 351 207 119 67 37 20
...
T(3, 1) = 2^2*(2*1+1) - 1 = 12 - 1 = 11. (End)
MAPLE
MATHEMATICA
Table[(2^# (2 k + 1)) - 1 &[m - k], {m, 0, 10}, {k, 0, m}] (* Michael De Vlieger, Jun 05 2016 *)
PROG
(PARI) T(n, k)= 2^n * (2*k + 1) - 1; \\ Ruud H.G. van Tol, Nov 29 2025
CROSSREFS
KEYWORD
AUTHOR
Antti Karttunen, Sep 12 2002
STATUS
approved