A379335 - OEIS

%I #12 Dec 23 2024 02:15:51

%S 1,2,4,8,16,48,96,192,384,768,1536,5632,11264,22528,77312,154624,

%T 309248,922624,1845248,6848512,17096576,34193152,68386304,272849152

%N Number of subsets of {-n..n} whose sum of reciprocals is 1.

%C Number of ways of writing 1 as Sum_{k=-n..n, k<>0} e(k)/k, where e(k) is 0 or 1.

%F a(n) <= 2*a(n-1) since we count s and s union {-1/n, 1/n} for each subset s counted in a(n-1); equality holds for n prime (and other cases). - _Michael S. Branicky_, Dec 21 2024

%e a(3) = 4 subsets: {1}, {-3, 1, 3}, {-2, 1, 2}, {-3, -2, 1, 2, 3}.

%o (Python)

%o from functools import cache

%o from fractions import Fraction

%o @cache

%o def b(i, s):

%o     if i == 0: return 1 if s == 1 else 0

%o     return b(i-1, s) + b(i-1, s+Fraction(1, (-1)**(i&1)*((i+1)>>1)))

%o a = lambda n: b(2*n, 0)

%o print([a(n) for n in range(1, 11)]) # _Michael S. Branicky_, Dec 21 2024

%Y Cf. A092670.

%K nonn,more

%O 1,2

%A _Ilya Gutkovskiy_, Dec 21 2024

%E a(12)-a(24) from _Michael S. Branicky_, Dec 21 2024