A358560 a(n) = Sum_{k=0..floor(n/3)} (n-k)!/(k! * (n-3*k)!).
1, 1, 1, 3, 7, 13, 33, 91, 223, 597, 1753, 4963, 14391, 44413, 137137, 427083, 1382383, 4534981, 14981673, 50719507, 174494983, 605276973, 2135204161, 7647369403, 27643067007, 101211363253, 375548195833, 1406858084931, 5326762882903, 20403498329437
Offset: 0
Links
- Seiichi Manyama, Table of n, a(n) for n = 0..1000
Programs
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PARI
a(n) = sum(k=0, n\3, (n-k)!/(k!*(n-3*k)!));
Formula
a(n) = (4 * a(n-1) - a(n-2) + 2 * (2*n-3) * a(n-3))/3 for n > 2.
a(n) ~ c * 2^(2*n/3) * n^(n/3) / (3^(n/3) * exp(n/3 - 2^(1/3) * n^(2/3) / 3^(2/3) + n^(1/3) / (2^(4/3) * 3^(7/3)))) * (1 + 7795/(5832*6^(2/3)*n^(1/3)) + 135724109/(2040733440*6^(1/3)*n^(2/3)) - 5962064767253/(42845606719488*n)), where c = 0.46562048925..., conjecture: c = sqrt(2) * exp(-1/81) / 3. - Vaclav Kotesovec, Nov 25 2022