A351995 - OEIS

A351995

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.

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

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OFFSET

0,7

COMMENTS

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

LINKS

Rémy Sigrist, Table of n, a(n) for n = 0..10010

Index entries for sequences related to binary expansion of n

FORMULA

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

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

A(n, 1) = n.

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

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

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

A(0, k) = 0.

A(1, k) = 1.

A(2, k) = 2^k.

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

EXAMPLE

Square array A(n, k) begins:

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

------------------------------------------------------------------

0| 0 0 0 0 0 0 0 0 0 0 0

1| 1 1 1 1 1 1 1 1 1 1 1

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

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

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

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

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

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

PROG

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

CROSSREFS

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

Sequence in context: A037836 A194522 A165013 * A055290 A125629 A339160

Adjacent sequences: A351992 A351993 A351994 * A351996 A351997 A351998

KEYWORD

nonn,base,tabl

AUTHOR

Rémy Sigrist, Feb 27 2022

STATUS

approved