OFFSET
1,7
COMMENTS
2*a(n-1) for n > 1 is the number of subsets of {1..n} that contain 4 even numbers. For example, for n=8, 2*a(7)=16 and the 16 subsets are {2,4,6,8}, {1,2,4,6,8}, {2,3,4,6,8}, {2,4,5,6,8}, {2,4,6,7,8}, {1,2,3,4,6,8}, {1,2,4,5,6,8}, {1,2,4,6,7,8}, {2,3,4,5,6,8}, {2,3,4,6,7,8}, {2,4,5,6,7,8}, {1,2,3,4,5,6,8}, {1,2,3,4,6,7,8}, {1,2,4,5,6,7,8}, {2,3,4,5,6,7,8}, {1,2,3,4,5,6,7,8}.
LINKS
Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (0,10,0,-40,0,80,0,-80,0,32).
FORMULA
a(n) = binomial((n+1)/2, 4) * 2^((n-1)/2), n odd;
a(n) = binomial((n/2), 4) * 2^(n/2), n even.
From Colin Barker, Jan 18 2020: (Start)
G.f.: 8*x^7*(1 + 2*x) / (1 - 2*x^2)^5.
a(n) = 10*a(n-2) - 40*a(n-4) + 80*a(n-6) - 80*a(n-8) + 32*a(n-10) for n>10. (End)
From Amiram Eldar, Mar 24 2022: (Start)
Sum_{n>=7} 1/a(n) = (5-6*log(2))/4.
Sum_{n>=7} (-1)^(n+1)/a(n) = (5-6*log(2))/12. (End)
EXAMPLE
a(7)=8 and the 8 subsets are {1,3,5,7}, {1,2,3,5,7}, {1,3,4,5,7}, {1,3,5,6,7}, {1,2,3,4,5,7}, {1,2,3,5,6,7}, {1,3,4,5,6,7}, {1,2,3,4,5,6,7}.
MATHEMATICA
a[n_] := If[OddQ[n], Binomial[(n + 1)/2, 4]*2^((n - 1)/2), Binomial[n/2, 4]*2^(n/2)]; Array[a, 38] (* Amiram Eldar, Jan 17 2020 *)
LinearRecurrence[{0, 10, 0, -40, 0, 80, 0, -80, 0, 32}, {0, 0, 0, 0, 0, 0, 8, 16, 80, 160}, 50] (* Harvey P. Dale, Jul 22 2024 *)
PROG
(Magma) [IsOdd(n) select Binomial((n+1) div 2, 4)*2^((n-1) div 2) else Binomial((n div 2), 4)*2^(n div 2): n in [1..38]]; // Marius A. Burtea, Jan 17 2020
(PARI) concat([0, 0, 0, 0, 0, 0], Vec(8*x^7*(1 + 2*x) / (1 - 2*x^2)^5 + O(x^40))) \\ Colin Barker, Jan 18 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Enrique Navarrete, Jan 16 2020
STATUS
approved