A372049 a(n) is the index of the Lucas number that is the ratio of the sum of the first n Fibonacci numbers divided by the largest possible Fibonacci number.

1, 1, 0, 4, 3, 3, 5, 6, 5, 5, 7, 8, 7, 7, 9, 10, 9, 9, 11, 12, 11, 11, 13, 14, 13, 13, 15, 16, 15, 15, 17, 18, 17, 17, 19, 20, 19, 19, 21, 22, 21, 21, 23, 24, 23, 23, 25, 26, 25, 25, 27, 28, 27, 27, 29, 30, 29, 29, 31, 32, 31, 31, 33, 34, 33, 33, 35, 36, 35, 35, 37, 38, 37, 37, 39, 40, 39, 39, 41, 42, 41

Offset: 1

Author

Tanya Khovanova and MIT PRIMES senior group, Apr 17 2024

Keywords

• nonn

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)+2.

Examples

			The sum of the first ten Fibonacci numbers is 143. The largest Fibonacci that divides this sum is 13, the seventh Fibonacci number. After the division we get 143/13 = 11, the fifth Lucas number. Thus, a(10) = 5.
		

Links

• Tanya Khovanova and the MIT PRIMES STEP senior group, Fibonacci Partial Sums Tricks, arXiv:2409.01296 [math.HO], 2024.

Crossrefs

Cf. A000032, A000045, A000071, A372048.