A059451 - OEIS

A059451

Number of ways n can be written as the sum of two numbers whose binary expansions have even numbers of zeros; also number of ways n can be written as the sum of two numbers whose binary expansions have odd numbers of zeros.

0, 1, 0, 1, 1, 1, 1, 2, 0, 2, 2, 1, 3, 2, 1, 3, 2, 2, 4, 3, 1, 4, 3, 3, 4, 4, 2, 4, 5, 3, 5, 5, 2, 5, 6, 3, 7, 6, 1, 7, 6, 4, 8, 6, 3, 7, 7, 5, 8, 7, 4, 9, 5, 6, 11, 6, 6, 9, 6, 7, 11, 8, 5, 10, 8, 7, 12, 8, 7, 11, 7, 9, 12, 10, 6, 12, 9, 7, 17, 9, 6, 13, 10, 9, 15, 12, 5, 14, 12, 9, 16, 11, 9, 14, 11

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,8

COMMENTS

The only place where the two sequences differ is a(0) which is 1 for the odds and 0 for the evens.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

IBM Ponder This, Feb. 2001

Y.-G. Chen and B. Wang, On additive properties of two special sequences, Acta Arith. 110 (3) (2003), 299-303.

EXAMPLE

a(16)=3 since 16=2+14=5+11=8+8 (in binary 10+1110=101+1011=1000+1000 where each term has an odd number of zeros) and since 16=1+15=4+12=7+9 (in binary 1+1111=100+1100=111+1001 where each term has an even number of zeros).

MAPLE

N:= 100: # for a(1)..a(N)

T:= select(t -> numboccur(0, convert(t, base, 2))::even, {$0..N}):

f:= proc(n) nops(map(t -> n-t, T intersect {$0..n/2}) intersect T) end proc:

map(f, [$1..N]); # Robert Israel, Nov 28 2024

PROG

(Python)

def c(n): return (n.bit_length() - n.bit_count())&1 == 0

def a(n): return sum(1 for i in range(1, n//2+1) if c(i) and c(n-i))

print([a(n) for n in range(1, 96)]) # Michael S. Branicky, Dec 02 2024

CROSSREFS

Cf. A059009 and A059010 for the odd and even binary zeros sequences.