A351995 Square array A(n, k), n, k >= 0, read by antidiagonals upwards; A(n, k) = Sum_{ i >= 0 } b_i * 2^(k*i) where n = Sum_{ i >= 0 } b_i * 2^i.

0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 1, 3, 4, 1, 0, 2, 4, 5, 8, 1, 0, 2, 5, 16, 9, 16, 1, 0, 3, 6, 17, 64, 17, 32, 1, 0, 1, 7, 20, 65, 256, 33, 64, 1, 0, 2, 8, 21, 72, 257, 1024, 65, 128, 1, 0, 2, 9, 64, 73, 272, 1025, 4096, 129, 256, 1, 0, 3, 10, 65, 512, 273, 1056, 4097, 16384, 257, 512, 1, 0

Offset: 0

Author

Rémy Sigrist, Feb 27 2022

Keywords

• nonn
• base
• tabl
• changed

Comments

In other words, in binary expansion of n, replace 2^i by 2^(k*i).

Examples

			Square array A(n, k) begins:
  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|  1  1   1   1    1     1     1      1      1       1        1
    2|  1  2   4   8   16    32    64    128    256     512     1024
    3|  2  3   5   9   17    33    65    129    257     513     1025
    4|  1  4  16  64  256  1024  4096  16384  65536  262144  1048576
    5|  2  5  17  65  257  1025  4097  16385  65537  262145  1048577
    6|  2  6  20  72  272  1056  4160  16512  65792  262656  1049600
    7|  3  7  21  73  273  1057  4161  16513  65793  262657  1049601
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..10010
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000120, A000695, A033045, A033052, A352001.

Programs

Mathematica

A351995[n_, k_] := If[n <= 1, n, Total[2^(k*(Flatten[Position[Reverse[IntegerDigits[n, 2]], 1]] - 1))]];
Table[A351995[n - k, k], {n, 0, 15}, {k, 0, n}] (* Paolo Xausa, Aug 26 2025 *)

PARI

A(n,k) = { my (v=0, e); while (n, n-=2^e=valuation(n, 2); v+=2^(k*e)); v }

Formula

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

A(n, 0) = A000120(n).

A(n, 1) = n.

A(n, 2) = A000695(n).

A(n, 3) = A033045(n).

A(n, 4) = A033052(n).

A(0, k) = 0.

A(1, k) = 1.

A(2, k) = 2^k.

A(3, k) = 2^k + 1.