A322403 Square array T(n, k) read by antidiagonals, n >= 0 and k >= 0: the lengths of runs in binary expansion of T(n, k) are obtained by multiplying those of n and of k (see Comments for precise definition).

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 2, 3, 0, 0, 4, 12, 12, 4, 0, 0, 5, 4, 15, 4, 5, 0, 0, 6, 42, 48, 48, 42, 6, 0, 0, 7, 6, 51, 16, 51, 6, 7, 0, 0, 8, 56, 60, 292, 292, 60, 56, 8, 0, 0, 9, 8, 63, 12, 5, 12, 63, 8, 9, 0, 0, 10, 150, 192, 448, 438, 438, 448, 192

Offset: 0

Author

Rémy Sigrist, Dec 06 2018

Keywords

• nonn
• base
• tabl

Comments

For any n >= 0 and k >= 0:

- let r_n be the lengths of runs in binary expansion of n,

- for n = 0: we assume that r_0 = (),

- when n > 0: let R_n be the #r_n-periodic sequence whose first #r_n terms match r_n,

- r_{T(n, k)} has lcm(#r_n, #r_k) terms and r_{T(n, k)}(i) = R_n(i) * R_k(i) for i = 1..lcm(#r_n, #r_k).

Examples

			Array T(n, k) begins (in decimal):
  n\k|  0  1   2    3    4     5    6     7     8     9     10
  ---+--------------------------------------------------------
    0|  0  0   0    0    0     0    0     0     0      0     0
    1|  0  1   2    3    4     5    6     7     8      9    10
    2|  0  2   2   12    4    42    6    56     8    150    10
    3|  0  3  12   15   48    51   60    63   192    195   204
    4|  0  4   4   48   16   292   12   448    64   2124    36
    5|  0  5  42   51  292     5  438   455  2184      9  2730
    6|  0  6   6   60   12   438   30   504    24   3294    54
    7|  0  7  56   63  448   455  504   511  3584   3591  3640
    8|  0  8   8  192   64  2184   24  3584   512  33048   136
Array T(n, k) begins (in binary):
   n\k|  0     1      10        11        100          101         110
  ----+---------------------------------------------------------------
     0|  0     0       0         0          0             0          0
     1|  0     1      10        11        100           101        110
    10|  0    10      10      1100        100        101010        110
    11|  0    11    1100      1111     110000        110011     111100
   100|  0   100     100    110000      10000     100100100       1100
   101|  0   101  101010    110011  100100100           101  110110110
   110|  0   110     110    111100       1100     110110110      11110
   111|  0   111  111000    111111  111000000     111000111  111111000
  1000|  0  1000    1000  11000000    1000000  100010001000      11000
		

Links

• Index entries for sequences related to binary expansion of n

Crossrefs

See A322404 for the additive variant.

Cf. A001196, A097254, A097262, A322149.

Programs

PARI

T(n,k) = my (v=0, p=1, rn=n, rk=k, b=if ((max(n,1)%2)&&(max(k,1)%2), 1, 0)); while (1, my (vn=if (rn==0, 0, valuation(rn+(rn%2), 2)), vk=if(rk==0, 0, valuation(rk+(rk%2), 2)), w=vn*vk); v+=b*p*(2^w-1); rn\=2^vn; rk\=2^vk; if (rn==0 && rk==0, return (v), rn==0, rn=n, rk==0, rk=k); p*=2^w; b=1-b)

Formula

For any m >= 0, n >= 0 and k >= 0:

- T(n, k) = T(k, n) (T is commutative),

- T(m, T(n, k)) = T(T(m, n), k) (T is associative),

- T(m, A322404(n, k)) = A322404(T(m, n), T(m, k)) (T distributes over A322404),

- T(n, 0) = 0 (0 is an absorbing element for T),

- T(n, 1) = n (1 is an neutral element for T),

- T(n, 3) = A001196(n),

- T(n, 7) = A097254(n+1),

- T(n, 15) = A097262(n),

- T(n, n) = A322149(n),

- A005811(T(n, k)) = lcm(A005811(n), A005811(k)),

- T(2^n - 1, 2^k - 1) = 2^(n*k) - 1.

- T(2^n, 2^k) = 2^(n*k) when n > 0 and k > 0,

- T(n, k) is odd iff both n and k are odd.