A351995 - OEIS

%I #15 Mar 01 2022 14:51:56

%S 0,1,0,1,1,0,2,2,1,0,1,3,4,1,0,2,4,5,8,1,0,2,5,16,9,16,1,0,3,6,17,64,

%T 17,32,1,0,1,7,20,65,256,33,64,1,0,2,8,21,72,257,1024,65,128,1,0,2,9,

%U 64,73,272,1025,4096,129,256,1,0,3,10,65,512,273,1056,4097,16384,257,512,1,0

%N Square array A(n, k), n, k >= 0, read by antidiagonals upwards; A(n, k) = Sum_{ i >= 0 } b_i * 2^(k*i) where n = Sum_{ i >= 0 } b_i * 2^i.

%C In other words, in binary expansion of n, replace 2^i by 2^(k*i).

%H Rémy Sigrist, <a href="/A351995/b351995.txt">Table of n, a(n) for n = 0..10010</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

%F A(A(n, k), k') = A(n, k*k') for k, k' > 0.

%F A(n, 0) = A000120(n).

%F A(n, 1) = n.

%F A(n, 2) = A000695(n).

%F A(n, 3) = A033045(n).

%F A(n, 4) = A033052(n).

%F A(0, k) = 0.

%F A(1, k) = 1.

%F A(2, k) = 2^k.

%F A(3, k) = 2^k + 1.

%e Square array A(n, k) begins:

%e   n\k|  0  1   2   3    4     5     6      7      8       9       10

%e   ------------------------------------------------------------------

%e     0|  0  0   0   0    0     0     0      0      0       0        0

%e     1|  1  1   1   1    1     1     1      1      1       1        1

%e     2|  1  2   4   8   16    32    64    128    256     512     1024

%e     3|  2  3   5   9   17    33    65    129    257     513     1025

%e     4|  1  4  16  64  256  1024  4096  16384  65536  262144  1048576

%e     5|  2  5  17  65  257  1025  4097  16385  65537  262145  1048577

%e     6|  2  6  20  72  272  1056  4160  16512  65792  262656  1049600

%e     7|  3  7  21  73  273  1057  4161  16513  65793  262657  1049601

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

%Y Cf. A000120, A000695, A033045, A033052, A352001.

%K nonn,base,tabl

%O 0,7

%A _Rémy Sigrist_, Feb 27 2022