OFFSET
1,5
COMMENTS
2*a(n-1) for n > 1 is the number of subsets of {1..n} that contain three even numbers. For example, for n=6, 2*a(5)=8 and the 8 subsets are {2,4,6}, {1,2,4,6}, {2,3,4,6}, {2,4,5,6}, {1,2,3,4,6}, {1,2,4,5,6}, {2,3,4,5,6}, {1,2,3,4,5,6}.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,8,0,-24,0,32,0,-16).
FORMULA
a(n) = binomial((n+1)/2,3) * 2^((n-1)/2), n odd;
a(n) = binomial(n/2,3) * 2^(n/2), n even.
From Colin Barker, Jan 17 2020: (Start)
G.f.: 4*x^5*(1 + 2*x) / (1 - 2*x^2)^4.
a(n) = 8*a(n-2) - 24*a(n-4) + 32*a(n-6) - 16*a(n-8) for n>8. (End)
From Amiram Eldar, Mar 24 2022: (Start)
Sum_{n>=5} 1/a(n) = (9/8)*(2*log(2)-1).
Sum_{n>=5} (-1)^(n+1)/a(n) = (3/8)*(2*log(2)-1). (End)
EXAMPLE
For n = 6, a(6) = 8 and the 8 subsets are {1,3,5}, {1,2,3,5}, {1,3,4,5}, {1,3,5,6}, {1,2,3,4,5}, {1,2,3,5,6}, {1,3,4,5,6}, {1,2,3,4,5,6}.
MATHEMATICA
a[n_] := If[OddQ[n], Binomial[(n + 1)/2, 3]*2^((n - 1)/2), Binomial[n/2, 3]*2^(n/2)]; Array[a, 39] (* Amiram Eldar, Jan 17 2020 *)
PROG
(PARI) concat([0, 0, 0, 0], Vec(4*x^5*(1 + 2*x) / (1 - 2*x^2)^4 + O(x^40))) \\ Colin Barker, Jan 17 2020
(Magma) [IsOdd(n) select Binomial((n+1) div 2, 3)*2^((n-1) div 2) else Binomial((n div 2), 3)*2^(n div 2): n in [1..39]]; // Marius A. Burtea, Jan 17 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Enrique Navarrete, Jan 16 2020
STATUS
approved