

A302437


a(n) is the number of ways of writing the binary expansion of n as a concatenation of nonempty substrings with Hamming weight at most 2.


2



1, 1, 2, 2, 4, 4, 4, 3, 8, 8, 8, 7, 8, 7, 6, 5, 16, 16, 16, 15, 16, 15, 14, 11, 16, 15, 14, 13, 12, 11, 10, 8, 32, 32, 32, 31, 32, 31, 30, 23, 32, 31, 30, 27, 28, 25, 22, 18, 32, 31, 30, 29, 28, 27, 26, 20, 24, 23, 22, 20, 20, 18, 16, 13, 64, 64, 64, 63, 64
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OFFSET

0,3


COMMENTS

Leading zeros in the binary expansion of n are ignored.
The value a(0) = 1 corresponds to the empty concatenation.
See A301453 for similar sequences.


LINKS

Rémy Sigrist, Table of n, a(n) for n = 0..10000


FORMULA

a(2^n  1) = A000045(n + 1) for any n >= 0.
a(2^n) = 2^n for any n >= 0.
a(2*n) = 2*a(n) for any n > 0.
If 1 <= A000120(n) <= 2 then a(n) = A053644(n).


EXAMPLE

For n = 14: the binary expansion of 14, "1110", can be split in 6 ways into substrings with Hamming weight at most 2:
 (11)(10),
 (11)(1)(0),
 (1)(110),
 (1)(11)(0),
 (1)(1)(10),
 (1)(1)(1)(0).
Hence a(14) = 6.


PROG

(PARI) a(n) = if (n==0, return (1), my (v=0, h=0); while (n, h += n%2; n\=2; if (h>2, break, v += a(n))); return (v))


CROSSREFS

Cf. A000045, A000120, A053644, A301453.
Sequence in context: A165956 A263991 A065285 * A179932 A267649 A071805
Adjacent sequences: A302434 A302435 A302436 * A302438 A302439 A302440


KEYWORD

nonn,base


AUTHOR

Rémy Sigrist, Apr 08 2018


STATUS

approved



