This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A380408 #9 Feb 05 2025 22:24:22
%S A380408 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,
%T A380408 38,40,41,43,44,46,47,49,50,52,53,55,56,58,59,61,62,64,65,67,68,70,71,
%U A380408 74,75,77,78,80,81,83,84,86,87,89,90,92,93,95,96,98,99,101,102,104,105,107
%N A380408 a(n) = Sum_{k>=0} floor(n/(2k)!).
%C A380408 Partial sum of A060832 except for the first term in the sum.
%C A380408 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.
%C A380408 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.
%F A380408 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.
%o A380408 (PARI) a(n) = round(sumpos(k=0, n\(2*k)!)); \\ _Michel Marcus_, Jan 24 2025
%Y A380408 Cf. A060832, A034968, A013936, A013939, A038663.
%K A380408 nonn
%O A380408 0,3
%A A380408 _Akiva Weinberger_, Jan 23 2025