login
Number of times that the numerator of a sum generated from the set 1, 1/2, 1/3,..., 1/n is a Fibonacci number.
4

%I #23 Feb 15 2022 11:38:22

%S 1,3,5,9,11,22,28,37,45,62,70,125,133,172,330,421,450,840,901,1710,

%T 2356,2724,2824,5367,6022,7142,8072,18771,19204,35739,36453,42853,

%U 82094,88574,155642,264869

%N Number of times that the numerator of a sum generated from the set 1, 1/2, 1/3,..., 1/n is a Fibonacci number.

%C Note that for each n there are only 2^(n-1) new sums to consider. For the number of distinct Fibonacci numbers, see A256221. For the largest generated Fibonacci number, see A256222. For the smallest Fibonacci number not generated, see A256223.

%e a(3) = 5 because we obtain 5 following subsets {1}, {1/2}, {1/3}, {1,1/2} and {1/2, 1/3} having 5 sums with Fibonacci numerators: 1, 1, 1, 1+1/2 = 3/2 and 1/2+1/3 = 5/6.

%t <<"DiscreteMath`Combinatorica`"; maxN=22; For[cnt=0; i=0; n=1, n<=maxN, n++, While[i<2^n-1, i++; s=NthSubset[i, Range[n]]; k=Numerator[Plus@@(1/s)]; If[IntegerQ[Sqrt[5*k^2+4]]||IntegerQ[Sqrt[5*k^2-4]], cnt++ ]]; Print[cnt]]

%o (Python)

%o from math import gcd, lcm

%o from itertools import combinations

%o def A256220(n):

%o m = lcm(*range(1,n+1))

%o fibset, mlist = set(), tuple(m//i for i in range(1,n+1))

%o a, b, c, k = 0, 1, 0, sum(mlist)

%o while b <= k:

%o fibset.add(b)

%o a, b = b, a+b

%o for l in range(1,n//2+1):

%o for p in combinations(mlist,l):

%o s = sum(p)

%o if s//gcd(s,m) in fibset:

%o c += 1

%o if 2*l != n and (k-s)//gcd(k-s,m) in fibset:

%o c += 1

%o return c+int(k//gcd(k,m) in fibset) # _Chai Wah Wu_, Feb 15 2022

%Y Cf. A000045, A075188, A010056, A256221, A256222, A256223.

%K nonn,more

%O 1,2

%A _Michel Lagneau_, Mar 19 2015

%E a(23)-a(36) from _Lars Blomberg_, May 06 2015