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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. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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.
LINKS
EXAMPLE
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.
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]]
PROG
(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
CROSSREFS
Sequence in context: A231716 A113488 A092917 * A163778 A328643 A160358
KEYWORD
nonn,more
AUTHOR
Michel Lagneau, Mar 19 2015
EXTENSIONS
a(23)-a(36) from Lars Blomberg, May 06 2015
STATUS
approved

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Last modified April 24 05:23 EDT 2024. Contains 371918 sequences. (Running on oeis4.)