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 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. 2
 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 (list; table; graph; refs; listen; history; text; internal format)
 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 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: A060572 A163543 A180009 * A180010 A287401 A003406 Adjacent sequences:  A341904 A341905 A341906 * A341908 A341909 A341910 KEYWORD nonn,tabl,base AUTHOR Rémy Sigrist, Jun 04 2021 STATUS approved

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Last modified August 1 22:01 EDT 2021. Contains 346408 sequences. (Running on oeis4.)