A341907 T(n, k) is the result of replacing 2^e with k^e in the binary expansion of n; square array T(n, k) read by antidiagonals upwards, n, k >= 0.

0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 0, 2, 2, 1, 0, 1, 1, 3, 3, 1, 0, 0, 2, 4, 4, 4, 1, 0, 1, 2, 5, 9, 5, 5, 1, 0, 0, 3, 6, 10, 16, 6, 6, 1, 0, 1, 1, 7, 12, 17, 25, 7, 7, 1, 0, 0, 2, 8, 13, 20, 26, 36, 8, 8, 1, 0, 1, 2, 9, 27, 21, 30, 37, 49, 9, 9, 1, 0, 0, 3, 10, 28, 64, 31, 42, 50, 64, 10, 10, 1, 0

Offset: 0

Author

Rémy Sigrist, Jun 04 2021

Keywords

• nonn
• tabl
• base

Comments

For any n >= 0, the n-th row, k -> T(n, k), corresponds to a polynomial in k with coefficients in {0, 1}.

For any k > 1, the k-th column, n -> T(n, k), corresponds to sums of distinct powers of k.

Examples

			Array T(n, k) begins:
  n\k|  0  1   2   3   4    5    6    7    8    9    10    11    12
  ---+-------------------------------------------------------------
    0|  0  0   0   0   0    0    0    0    0    0     0     0     0
    1|  1  1   1   1   1    1    1    1    1    1     1     1     1
    2|  0  1   2   3   4    5    6    7    8    9    10    11    12
    3|  1  2   3   4   5    6    7    8    9   10    11    12    13
    4|  0  1   4   9  16   25   36   49   64   81   100   121   144
    5|  1  2   5  10  17   26   37   50   65   82   101   122   145
    6|  0  2   6  12  20   30   42   56   72   90   110   132   156
    7|  1  3   7  13  21   31   43   57   73   91   111   133   157
    8|  0  1   8  27  64  125  216  343  512  729  1000  1331  1728
    9|  1  2   9  28  65  126  217  344  513  730  1001  1332  1729
   10|  0  2  10  30  68  130  222  350  520  738  1010  1342  1740
   11|  1  3  11  31  69  131  223  351  521  739  1011  1343  1741
   12|  0  2  12  36  80  150  252  392  576  810  1100  1452  1872
		

Links

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

Crossrefs

See A342707 for a similar sequence.

Cf. A000035, A000120, A000583, A000695, A001093, A002061, A002378, A002522, A002523, A005836, A007088, A011379, A027444, A033042, A033043, A033044, A033045, A033046, A033047, A033048, A033049, A034262, A053698, A071568, A098547, A104258.

Programs

PARI

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

Formula

T(n, n) = A104258(n).

T(n, 0) = A000035(n).

T(n, 1) = A000120(n).

T(n, 2) = n.

T(n, 3) = A005836(n).

T(n, 4) = A000695(n).

T(n, 5) = A033042(n).

T(n, 6) = A033043(n).

T(n, 7) = A033044(n).

T(n, 8) = A033045(n).

T(n, 9) = A033046(n).

T(n, 10) = A007088(n).

T(n, 11) = A033047(n).

T(n, 12) = A033048(n).

T(n, 13) = A033049(n).

T(0, k) = 0.

T(1, k) = 1.

T(2, k) = k.

T(3, k) = k + 1.

T(4, k) = k^2.

T(5, k) = k^2 + 1 = A002522(k).

T(6, k) = k^2 + k = A002378(k).

T(7, k) = k^2 + k + 1 = A002061(k).

T(8, k) = k^3.

T(9, k) = k^3 + 1 = A001093(k).

T(10, k) = k^3 + k = A034262(k).

T(11, k) = k^3 + k + 1 = A071568(k).

T(12, k) = k^3 + k^2 = A011379(k).

T(13, k) = k^3 + k^2 + 1 = A098547(k).

T(14, k) = k^3 + k^2 + k = A027444(k).

T(15, k) = k^3 + k^2 + k + 1 = A053698(k).

T(16, k) = k^4 = A000583(k).

T(17, k) = k^4 + 1 = A002523(k).

T(m + n, k) = T(m, k) + T(n, k) when m AND n = 0 (where AND denotes the bitwise AND operator).