A351785 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, 1, 1, 2, 4, 2, 3, 3, 3, 3, 4, 6, 1, 6, 4, 5, 5, 4, 4, 5, 5, 6, 16, 6, 5, 6, 16, 6, 7, 7, 7, 7, 7, 7, 7, 7, 8, 18, 5, 18, 16, 18, 5, 18, 8, 9, 9, 16, 16, 17, 17, 16, 16, 9, 9, 10, 12, 10, 17, 18, 20, 18, 17, 10, 12, 10, 11, 11, 11, 11, 19, 19, 19, 19, 11, 11, 11, 11

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   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15
    1|   1   4   3   6   5  16   7  18   9  12  11  14  13  24  15  26
    2|   2   3   1   4   6   7   5  16  10  11   9  12  14  15  13  24
    3|   3   6   4   5   7  18  16  17  11  14  12  13  15  26  24  25
    4|   4   5   6   7  16  17  18  19  12  13  14  15  24  25  26  27
    5|   5  16   7  18  17  20  19  22  13  24  15  26  25  28  27  30
    6|   6   7   5  16  18  19  17  20  14  15  13  24  26  27  25  28
    7|   7  18  16  17  19  22  20  21  15  26  24  25  27  30  28  29
    8|   8   9  10  11  12  13  14  15   2   3   1   4   6   7   5  16
    9|   9  12  11  14  13  24  15  26   3   6   4   5   7  18  16  17
   10|  10  11   9  12  14  15  13  24   1   4   3   6   5  16   7  18
		

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, A351786 (multiplication).

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) = n.