A075300 Array A read by antidiagonals upwards: A(n, k) = array A054582(n,k) - 1 = 2^n*(2*k+1) - 1 with n,k >= 0.

0, 1, 2, 3, 5, 4, 7, 11, 9, 6, 15, 23, 19, 13, 8, 31, 47, 39, 27, 17, 10, 63, 95, 79, 55, 35, 21, 12, 127, 191, 159, 111, 71, 43, 25, 14, 255, 383, 319, 223, 143, 87, 51, 29, 16, 511, 767, 639, 447, 287, 175, 103, 59, 33, 18, 1023, 1535, 1279, 895, 575, 351, 207, 119

Offset: 0

Author

Antti Karttunen, Sep 12 2002

Keywords

• nonn
• tabl
• easy

Comments

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) = A238087(n),

A(n,8) = A198275(n),

A(n,9) = A198276(n),

A(n,10) = A171389(n). (End)

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

Examples

			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)
		

Links

• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Inverse permutation: A075301. Transpose: A075302. The X-projection is given by A007814(n+1) and the Y-projection A025480.

Cf. A002262, A025581, A054582, A241957.

Programs

Maple

A075300bi := (x,y) -> (2^x * (2*y + 1))-1;
A075300 := n -> A075300bi(A025581(n), A002262(n));
A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);
A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))),2) - (n+1);

Mathematica

Table[(2^# (2 k + 1)) - 1 &[m - k], {m, 0, 10}, {k, 0, m}] (* Michael De Vlieger, Jun 05 2016 *)

Formula

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)