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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.

%I A340666 #27 Feb 03 2021 17:05:57
%S A340666 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,
%T A340666 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,
%U A340666 4096,65,48,7,64,9,3,0,1,512,3,16384,129,96,7,512,33,10,7
%N 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.
%H A340666 Alois P. Heinz, <a href="/A340666/b340666.txt">Antidiagonals n = 0..200, flattened</a>
%F A340666 A000120(A(n,k)) = A000120(n) = log_2(A(n,0)+1).
%F A340666 A023416(A(n,k)) = k * A023416(n) for n >= 1.
%e A340666 Square array A(n,k) begins:
%e A340666   0, 0,  0,   0,    0,     0,      0,       0,        0, ...
%e A340666   1, 1,  1,   1,    1,     1,      1,       1,        1, ...
%e A340666   1, 2,  4,   8,   16,    32,     64,     128,      256, ...
%e A340666   3, 3,  3,   3,    3,     3,      3,       3,        3, ...
%e A340666   1, 4, 16,  64,  256,  1024,   4096,   16384,    65536, ...
%e A340666   3, 5,  9,  17,   33,    65,    129,     257,      513, ...
%e A340666   3, 6, 12,  24,   48,    96,    192,     384,      768, ...
%e A340666   7, 7,  7,   7,    7,     7,      7,       7,        7, ...
%e A340666   1, 8, 64, 512, 4096, 32768, 262144, 2097152, 16777216, ...
%e A340666   ...
%p A340666 A:= (n, k)-> Bits[Join](subs(0=[0$k][], Bits[Split](n))):
%p A340666 seq(seq(A(n, d-n), n=0..d), d=0..12);
%p A340666 # second Maple program:
%p A340666 A:= proc(n, k) option remember; `if`(n<2, n,
%p A340666      `if`(irem(n, 2, 'r')=1, A(r, k)*2+1, A(r, k)*2^k))
%p A340666     end:
%p A340666 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A340666 A[n_, k_] := FromDigits[IntegerDigits[n, 2] /. 0 -> Sequence @@ Table[0, {k}], 2];
%t A340666 Table[A[n, d-n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Feb 02 2021 *)
%Y A340666 Columns k=0-2, 4 give: A038573, A001477, A084471, A084473.
%Y A340666 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).
%Y A340666 Main diagonal gives A340667.
%Y A340666 Cf. A000120, A023416.
%K A340666 nonn,tabl,look,base
%O A340666 0,9
%A A340666 _Alois P. Heinz_, Jan 15 2021