A351786 Symmetric array T(n, k), n, k >= 0, read by antidiagonals; for any m >= 0 with binary expansion Sum_{i >= 0} b_i*2^i, let d(m) = Sum_{i >= 0} b_i * 2^A130472(i); let t be the inverse of d; T(n, k) = t(d(n) * d(k)).

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 8, 3, 0, 0, 4, 10, 10, 4, 0, 0, 5, 1, 12, 1, 5, 0, 0, 6, 3, 5, 5, 3, 6, 0, 0, 7, 9, 18, 16, 18, 9, 7, 0, 0, 8, 11, 15, 20, 20, 15, 11, 8, 0, 0, 9, 32, 25, 17, 65, 17, 25, 32, 9, 0, 0, 10, 34, 40, 21, 23, 23, 21, 40, 34, 10, 0

Offset: 0

Author

Rémy Sigrist, Feb 19 2022

Keywords

• nonn
• base
• tabl

Comments

The function d is a bijection from the nonnegative integers to the nonnegative dyadic rationals satisfying d(A000695(n)) = n for any n >= 0.

Examples

			Array T(n, k) begins:
  n\k|  0   1   2   3   4   5    6    7    8    9   10   11   12   13   14   15
  ---+-------------------------------------------------------------------------
    0|  0   0   0   0   0   0    0    0    0    0    0    0    0    0    0    0
    1|  0   1   2   3   4   5    6    7    8    9   10   11   12   13   14   15
    2|  0   2   8  10   1   3    9   11   32   34   40   42   33   35   41   43
    3|  0   3  10  12   5  18   15   25   40   43   33   38   45   58   48   51
    4|  0   4   1   5  16  20   17   21    2    6    3    7   18   22   19   23
    5|  0   5   3  18  20  65   23   70   10   15   12   25   30   75   72   77
    6|  0   6   9  15  17  23   28   74   34   37   43   56   51   96   62  105
    7|  0   7  11  25  21  70   74   88   42   56   38   52   63  109   99  113
    8|  0   8  32  40   2  10   34   42  128  136  160  168  130  138  162  170
    9|  0   9  34  43   6  15   37   56  136  131  170  164  142  144  173  178
   10|  0  10  40  33   3  12   43   38  160  170  130  137  163  172  132  142
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10010
• Rémy Sigrist, Colored representation of the table for n, k < 2^10 (where the hue is function of T(n, k))
• Wikipedia, Dyadic rational
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000695, A130472, A351705, A351706, A351785 (addition).

Programs

PARI

d(n) = { my (v=0, k); while (n, n-=2^k=valuation(n, 2); v+=2^((-1)^k*(k+1)\2)); v }
t(n) = { my (v=0, k); while (n, n-=2^k=valuation(n, 2); v+=2^if (k>=0, 2*k, -1-2*k)); v }
T(n,k) = t(d(n)*d(k))

Formula

T(A000695(n), A000695(k)) = A000695(n * k).

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

T(m, T(n, k)) = T(T(m, n), k).

T(n, 0) = 0.

T(n, 1) = n.