A059451 - OEIS

%I #22 Dec 03 2024 12:29:42

%S 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,

%T 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,

%U 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

%N 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.

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

%H Robert Israel, <a href="/A059451/b059451.txt">Table of n, a(n) for n = 1..10000</a>

%H IBM Ponder This, <a href="https://research.ibm.com/haifa/ponderthis/challenges/February2001.html">Feb. 2001</a>

%H Y.-G. Chen and B. Wang, <a href="https://www.impan.pl/en/publishing-house/journals-and-series/acta-arithmetica/all/110/3">On additive properties of two special sequences</a>, Acta Arith. 110 (3) (2003), 299-303.

%e 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).

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

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

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

%p map(f, [$1..N]); # _Robert Israel_, Nov 28 2024

%o (Python)

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

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

%o print([a(n) for n in range(1, 96)]) # _Michael S. Branicky_, Dec 02 2024

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

%K nonn

%O 1,8

%A _Henry Bottomley_, Feb 02 2001