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

0,12

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.

Links

Rémy Sigrist, Table of n, a(n) for n = 0..10010

Index entries for sequences related to binary expansion of n

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).

Example

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

Prog

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

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.

Sequence in context: A163543 A358095 A180009 * A180010 A287401 A003406

Adjacent sequences: A341904 A341905 A341906 * A341908 A341909 A341910

Keyword

nonn,tabl,base

Author

Rémy Sigrist, Jun 04 2021

Status

approved