A355090 Square array A(n, k), n >= 0, k > 0, read by antidiagonals upwards; A(n, k) is the unique m such that n/k = fusc(m)/fusc(m+1) (where fusc is Stern's diatomic series A002487).

0, 1, 0, 3, 2, 0, 7, 1, 4, 0, 15, 5, 6, 8, 0, 31, 3, 1, 2, 16, 0, 63, 11, 9, 14, 12, 32, 0, 127, 7, 13, 1, 10, 4, 64, 0, 255, 23, 3, 17, 30, 2, 24, 128, 0, 511, 15, 19, 5, 1, 6, 28, 8, 256, 0, 1023, 47, 27, 29, 33, 62, 18, 20, 48, 512, 0, 2047, 31, 7, 3, 25, 1, 22, 2, 4, 16, 1024, 0

Offset: 0

Author

Rémy Sigrist, Jun 18 2022

Keywords

• nonn
• tabl

Comments

The binary expansion of A(n, k) encodes the position of n/k (> 0) in the Calkin-Wilf tree.

Examples

			Square array A(n, k) begins:
  n\k|      1    2    3    4    5    6    7    8    9    10    11    12
  ---+-----------------------------------------------------------------
    0|      0    0    0    0    0    0    0    0    0     0     0     0
    1|      1    2    4    8   16   32   64  128  256   512  1024  2048
    2|      3    1    6    2   12    4   24    8   48    16    96    32
    3|      7    5    1   14   10    2   28   20    4    56    40     8
    4|     15    3    9    1   30    6   18    2   60    12    36     4
    5|     31   11   13   17    1   62   22   26   34     2   124    44
    6|     63    7    3    5   33    1  126   14    6    10    66     2
    7|    127   23   19   29   25   65    1  254   46    38    58    50
    8|    255   15   27    3   21    9  129    1  510    30    54     6
    9|    511   47    7   35   61    5   49  257    1  1022    94    14
   10|   1023   31   39   11    3   13   57   17  513     1  2046    62
   11|   2047   95   55   59   67  125   37   41   97  1025     1  4094
   12|   4095   63   15    7   51    3   45    5    9    33  2049     1
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10152
• Wikipedia, Calkin-Wilf tree
• Index entries for sequences related to Stern's sequences

Crossrefs

Cf. A000918, A002487, A054429.

Programs

PARI

A(x,y) = { if (x==0, 0, my (v=0,t=1,a=0,b=1,c=1,d=0); while (1, my (m=a+c, n=b+d); if (x*n==y*m, return (t+v), x*n
				

Formula

A(m*n, m*k) = A(n, k) for any m > 0.

A(k, n) = A054429(A(n, k)) for any n, k > 0.

A(0, k) = 0.

A(1, k) = 2^(k-1).

A(n, 1) = 2^n - 1.

A(n, n+1) = A000918(n+1).

A(A002487(n), A002487(n+1)) = n.