A322261 Square array T(n, k) (n >= 0, k >= 0) read by antidiagonals upwards: the lengths of runs in binary expansion of T(n, k) correspond to the lengths of runs in binary expansion of n followed by the lengths of runs in binary expansion of k.

0, 1, 1, 2, 2, 2, 3, 5, 5, 3, 4, 6, 10, 4, 4, 5, 9, 13, 11, 11, 5, 6, 10, 18, 12, 20, 10, 6, 7, 13, 21, 19, 27, 21, 9, 7, 8, 14, 26, 20, 36, 26, 22, 8, 8, 9, 17, 29, 27, 43, 37, 25, 23, 23, 9, 10, 18, 34, 28, 52, 42, 38, 24, 40, 22, 10, 11, 21, 37, 35, 59, 53

Offset: 0

Author

Rémy Sigrist, Dec 01 2018

Keywords

• nonn
• tabl
• base

Comments

The array T is associative.

Examples

			Array T(n, k) begins (in decimal):
  n\k|  0   1   2   3   4   5   6   7    8    9   10   11   12
  ---+--------------------------------------------------------
    0|  0   1   2   3   4   5   6   7    8    9   10   11   12
    1|  1   2   5   4  11  10   9   8   23   22   21   20   19
    2|  2   5  10  11  20  21  22  23   40   41   42   43   44
    3|  3   6  13  12  27  26  25  24   55   54   53   52   51
    4|  4   9  18  19  36  37  38  39   72   73   74   75   76
    5|  5  10  21  20  43  42  41  40   87   86   85   84   83
    6|  6  13  26  27  52  53  54  55  104  105  106  107  108
    7|  7  14  29  28  59  58  57  56  119  118  117  116  115
    8|  8  17  34  35  68  69  70  71  136  137  138  139  140
Array T(n, k) begins (in binary):
  n\k |     0      1      10      11      100      101      110      111      1000
  ----+---------------------------------------------------------------------------
     0|     0      1      10      11      100      101      110      111      1000
     1|     1     10     101     100     1011     1010     1001     1000     10111
    10|    10    101    1010    1011    10100    10101    10110    10111    101000
    11|    11    110    1101    1100    11011    11010    11001    11000    110111
   100|   100   1001   10010   10011   100100   100101   100110   100111   1001000
   101|   101   1010   10101   10100   101011   101010   101001   101000   1010111
   110|   110   1101   11010   11011   110100   110101   110110   110111   1101000
   111|   111   1110   11101   11100   111011   111010   111001   111000   1110111
  1000|  1000  10001  100010  100011  1000100  1000101  1000110  1000111  10001000
		

Links

• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A004757, A005811, A010078, A020330, A042963, A047457, A047617, A101211, A163621.

Programs

PARI

torl(n) = my (r=[]); while (n, r = concat(valuation(n+(n%2),2), r); n \= 2^r[1];); r
fromrl(r) = my (v=0); for (i=1, #r, v = (v + (i%2))*2^r[i]-(i%2)); v
T(n,k) = fromrl(concat(torl(n), torl(k)))

Formula

T(n, 0) = T(0, n) = n.

T(n, 1) = A042963(n+1).

T(n, 2) = A047617(n+1).

T(n, 3) = A047457(n+1).

T(1, n) = A010078(n+1).

T(2, n) = A004757(n) for any n > 0.

A005811(T(n, k)) = A005811(n) + A005811(k).

T(2*n, k) = A163621(2*n, k) for any n > 0 and k > 0.

T(2*n, 2*n) = A020330(2*n) for any n > 0.