A380408 a(n) = Sum_{k>=0} floor(n/(2k)!).

0, 1, 3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 27, 28, 30, 31, 33, 34, 37, 38, 40, 41, 43, 44, 46, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 74, 75, 77, 78, 80, 81, 83, 84, 86, 87, 89, 90, 92, 93, 95, 96, 98, 99, 101, 102, 104, 105, 107

Offset

0,3

Comments

Partial sum of A060832 except for the first term in the sum.

Congruent to A034968(n) mod 2. Therefore, the parity of a(n) is the parity of the n-th permutation of k elements (k>=n) in lexicographic order.

For even n, a(n) equals A059563(n/2) whenever cosh(1)*n - a(n) < 1. The first time this fails is n=70, as a(70)=107 but A059563(35)=108. For small n, such failures appear to be very rare; however, the asymptotic density of these failures approaches 1.

Links

Table of n, a(n) for n=0..70.

Formula

a(n) = cosh(1)*n - f(n) where f(n) = Sum_{k>=0} fract(n/(2k)!). Here, fract() is the fractional part. The error term f(n) is unbounded above, and the greatest lower bound is 0 (even excluding n=0). The first values for which f(n) > s for s=1,2,3 are f(13)=1.06005, f(407) = 2.03382, and f(22319) = 3.01669. The error is almost periodic: for large m, f(n) is approximately f(n+(2m)!). If n is odd, f(n) > 1/2. f(n) alternately rises and descends, that is, f(2*n)<f(2*n+1)>f(2*n+2) for all n.

Prog

(PARI) a(n) = round(sumpos(k=0, n\(2*k)!)); \\ Michel Marcus, Jan 24 2025

Crossrefs

Cf. A060832, A034968, A013936, A013939, A038663.

Sequence in context: A049624 A084056 A032766 * A189935 A329987 A329962

Adjacent sequences: A380405 A380406 A380407 * A380409 A380410 A380411

Keyword

nonn,new

Author

Akiva Weinberger, Jan 23 2025

Status

approved