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A340666
A(n,k) is derived from n by replacing each 0 in its binary representation with a string of k 0's; square array A(n,k), n>=0, k>=0, read by antidiagonals.
5
0, 0, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 4, 3, 1, 0, 1, 8, 3, 4, 3, 0, 1, 16, 3, 16, 5, 3, 0, 1, 32, 3, 64, 9, 6, 7, 0, 1, 64, 3, 256, 17, 12, 7, 1, 0, 1, 128, 3, 1024, 33, 24, 7, 8, 3, 0, 1, 256, 3, 4096, 65, 48, 7, 64, 9, 3, 0, 1, 512, 3, 16384, 129, 96, 7, 512, 33, 10, 7
OFFSET
0,9
LINKS
FORMULA
A000120(A(n,k)) = A000120(n) = log_2(A(n,0)+1).
A023416(A(n,k)) = k * A023416(n) for n >= 1.
EXAMPLE
Square array A(n,k) begins:
0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 2, 4, 8, 16, 32, 64, 128, 256, ...
3, 3, 3, 3, 3, 3, 3, 3, 3, ...
1, 4, 16, 64, 256, 1024, 4096, 16384, 65536, ...
3, 5, 9, 17, 33, 65, 129, 257, 513, ...
3, 6, 12, 24, 48, 96, 192, 384, 768, ...
7, 7, 7, 7, 7, 7, 7, 7, 7, ...
1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, ...
...
MAPLE
A:= (n, k)-> Bits[Join](subs(0=[0$k][], Bits[Split](n))):
seq(seq(A(n, d-n), n=0..d), d=0..12);
# second Maple program:
A:= proc(n, k) option remember; `if`(n<2, n,
`if`(irem(n, 2, 'r')=1, A(r, k)*2+1, A(r, k)*2^k))
end:
seq(seq(A(n, d-n), n=0..d), d=0..12);
MATHEMATICA
A[n_, k_] := FromDigits[IntegerDigits[n, 2] /. 0 -> Sequence @@ Table[0, {k}], 2];
Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Feb 02 2021 *)
CROSSREFS
Columns k=0-2, 4 give: A038573, A001477, A084471, A084473.
Rows n=0..17, 19 give: A000004, A000012, A000079, A010701, A000302, A000051(k+1), A007283, A010727, A001018, A087289, A007582(k+1), A062709(k+2), A164346, A181565(k+1), A005009, A181404(k+3), A001025, A199493, A253208(k+1).
Main diagonal gives A340667.
Sequence in context: A320354 A285320 A347710 * A168068 A163575 A355889
KEYWORD
nonn,tabl,look,base
AUTHOR
Alois P. Heinz, Jan 15 2021
STATUS
approved