A356875 Square array, n >= 0, k >= 0, read by descending antidiagonals. A(n,k) = A022341(n)*2^k.

1, 2, 5, 4, 10, 9, 8, 20, 18, 17, 16, 40, 36, 34, 21, 32, 80, 72, 68, 42, 33, 64, 160, 144, 136, 84, 66, 37, 128, 320, 288, 272, 168, 132, 74, 41, 256, 640, 576, 544, 336, 264, 148, 82, 65, 512, 1280, 1152, 1088, 672, 528, 296, 164, 130, 69, 1024, 2560, 2304, 2176, 1344, 1056, 592, 328, 260, 138, 73

Offset: 0

Author

Peter Munn, Sep 02 2022

Keywords

• nonn
• easy
• tabl

Comments

The nonzero Fibbinary numbers (A003714) arranged in rows where each successive term is twice the preceding term; a (transposed) Fibbinary equivalent of A054582.

Write the first term in each row as Sum_{i in S} 2^i, where S is a set of nonnegative integers, then n = Sum_{i in S} F_i, where F_i is the i-th Fibonacci number, A000045(i).

More generally, if the terms are represented in binary, and the binary weighting of the digits (2^0, 2^1, 2^2, ...) is replaced with Fibonacci weighting (F_0, F_1, F_2, ...), we get the extended Wythoff array (A287870). If the weighting of the Zeckendorf representation is used (F_2, F_3, F_4, ...), we get the (unextended) Wythoff array (A035513).

Examples

			Square array A(n,k) begins:
   1    2    4    8    16    32    64   128 ...
   5   10   20   40    80   160   320   640 ...
   9   18   36   72   144   288   576  1152 ...
  17   34   68  136   272   544  1088  2176 ...
  21   42   84  168   336   672  1344  2688 ...
  33   66  132  264   528  1056  2112  4224 ...
  37   74  148  296   592  1184  2368  4736 ...
  41   82  164  328   656  1312  2624  5248 ...
  65  130  260  520  1040  2080  4160  8320 ...
  69  138  276  552  1104  2208  4416  8832 ...
  ...
The defining characteristic of a Fibbinary number is that its binary representation does not have a 1 followed by another 1. Shown in binary the array begins:
      1      10      100      1000 ...
    101    1010    10100    101000 ...
   1001   10010   100100   1001000 ...
  10001  100010  1000100  10001000 ...
  10101  101010  1010100  10101000 ...
  ...
		

Crossrefs

See the comments for the relationship to: A000045, A003714, A035513, A054582, A287870.

See the formula section for the relationship to: A022290, A022341, A356874.

Formula

A(n,0) = A022341(n), otherwise A(n,k) = 2*A(n,k-1).

A287870(n+1,k+1) = A356874(floor(A(n,k)/2)).

A035513(n+1,k+1) = A022290(A(n,k)).