OFFSET
1,1
COMMENTS
The sum of the first n Fibonacci numbers is sequence A000071.
When we divide the sum by the largest Fibonacci number that divides the sum, we always get a Lucas number.
For n > 3, a(n+4) = a(n)+1.
LINKS
Tanya Khovanova and the MIT PRIMES STEP senior group, Fibonacci Partial Sums Tricks, arXiv:2409.01296 [math.HO], 2024.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).
EXAMPLE
The sum of the first ten Fibonacci numbers is 143. The largest Fibonacci that divides this sum is 13, the seventh Fibonacci number. Thus, a(10) = 7. After the division we get 143/13 = 11, the fifth Lucas number.
MATHEMATICA
LinearRecurrence[{2, -2, 2, -1}, {2, 3, 2, 2}, 80] (* James C. McMahon, Apr 30 2024 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Tanya Khovanova and the MIT PRIMES STEP senior group, Apr 17 2024
STATUS
approved