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A342707
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T(n, k) is the result of replacing 2's by k's in the hereditary base-2 expansion of n; square array T(n, k) read by antidiagonals upwards, n, k >= 0.
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3
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0, 1, 0, 0, 1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 0, 2, 1, 3, 3, 1, 0, 1, 2, 4, 4, 4, 1, 0, 2, 2, 5, 27, 5, 5, 1, 0, 0, 3, 6, 28, 256, 6, 6, 1, 0, 1, 1, 7, 30, 257, 3125, 7, 7, 1, 0, 0, 2, 8, 31, 260, 3126, 46656, 8, 8, 1, 0, 1, 2, 9, 81, 261, 3130, 46657, 823543, 9, 9, 1, 0
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OFFSET
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0,12
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LINKS
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FORMULA
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T(n, 2) = n.
T(0, k) = 0.
T(1, k) = 1.
T(2, k) = k.
T(3, k) = k + 1.
T(7, k) = k^k + k + 1 = A066279(k).
T(m + n, k) = T(m, k) + T(n, k) when m AND n = 0 (where AND denotes the bitwise AND operator).
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EXAMPLE
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Array T(n, k) begins:
n\k| 0 1 2 3 4 5 6 7 8 9
---+-------------------------------------------------------------------
0| 0 0 0 0 0 0 0 0 0 0
1| 1 1 1 1 1 1 1 1 1 1
2| 0 1 2 3 4 5 6 7 8 9
3| 1 2 3 4 5 6 7 8 9 10
4| 1 1 4 27 256 3125 46656 823543 16777216 387420489
5| 2 2 5 28 257 3126 46657 823544 16777217 387420490
6| 1 2 6 30 260 3130 46662 823550 16777224 387420498
7| 2 3 7 31 261 3131 46663 823551 16777225 387420499
8| 0 1 8 81 1024 15625 279936 5764801 134217728 3486784401
9| 1 2 9 82 1025 15626 279937 5764802 134217729 3486784402
10| 0 2 10 84 1028 15630 279942 5764808 134217736 3486784410
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PROG
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(PARI) T(n, k) = { my (v=0, e); while (n, n-=2^e=valuation(n, 2); v+=k^T(e, k)); v }
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CROSSREFS
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See A341907 for a similar sequence.
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KEYWORD
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AUTHOR
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STATUS
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approved
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