A256220 Number of times that the numerator of a sum generated from the set 1, 1/2, 1/3,..., 1/n is a Fibonacci number.
1, 3, 5, 9, 11, 22, 28, 37, 45, 62, 70, 125, 133, 172, 330, 421, 450, 840, 901, 1710, 2356, 2724, 2824, 5367, 6022, 7142, 8072, 18771, 19204, 35739, 36453, 42853, 82094, 88574, 155642, 264869
Offset: 1
Examples
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.
Programs
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Mathematica
<<"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]]
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Python
from math import gcd, lcm from itertools import combinations def A256220(n): m = lcm(*range(1,n+1)) fibset, mlist = set(), tuple(m//i for i in range(1,n+1)) a, b, c, k = 0, 1, 0, sum(mlist) while b <= k: fibset.add(b) a, b = b, a+b for l in range(1,n//2+1): for p in combinations(mlist,l): s = sum(p) if s//gcd(s,m) in fibset: c += 1 if 2*l != n and (k-s)//gcd(k-s,m) in fibset: c += 1 return c+int(k//gcd(k,m) in fibset) # Chai Wah Wu, Feb 15 2022
Extensions
a(23)-a(36) from Lars Blomberg, May 06 2015
Comments