A373893 a(n) is the length of the simple continued fraction for the n-th alternating harmonic number.

0, 1, 2, 4, 7, 7, 5, 9, 8, 12, 9, 12, 11, 12, 13, 12, 18, 12, 17, 15, 15, 15, 19, 21, 18, 13, 21, 23, 25, 23, 26, 32, 28, 25, 24, 24, 31, 32, 33, 36, 41, 38, 38, 37, 44, 41, 37, 39, 47, 48, 42, 43, 43, 44, 46, 42, 44, 51, 45, 49, 52, 53, 62, 50, 57, 48, 55, 60, 52, 58, 70, 58, 60, 73, 67

Offset: 1

Author

Ilya Gutkovskiy, Jun 21 2024

Keywords

• nonn

Comments

By "simple continued fraction" is meant a continued fraction whose terms are positive integers and the final term is >= 2.

Examples

			Sum_{k=1..7} (-1)^(k+1)/k = 319/420 = 1/(1 + 1/(3 + 1/(6 + 1/(3 + 1/5)))), so a(7) = 5.
		

Crossrefs

Cf. A055573, A058312, A058313.

Programs

Mathematica

Table[Length[ContinuedFraction[Sum[(-1)^(k + 1)/k, {k, 1, n}]]] - 1, {n, 1, 75}]

Python

from fractions import Fraction
from sympy.ntheory.continued_fraction import continued_fraction
def A373893(n): return len(continued_fraction(sum(Fraction(1 if k&1 else -1,k) for k in range(1,n+1))))-1 # Chai Wah Wu, Jun 27 2024