OFFSET
0,5
COMMENTS
The general formula for the number of subsets of {1..n} that contain exactly k odd and j even numbers is binomial(ceiling(n/2), k) * binomial(floor(n/2), j).
LINKS
Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,-3,-3,3,1,-1).
FORMULA
a(n) = ceiling(n/2) * binomial(floor(n/2), 2).
From Colin Barker, Mar 01 2020: (Start)
G.f.: x^4*(2 + x) / ((1 - x)^4*(1 + x)^3).
a(n) = a(n-1) + 3*a(n-2) - 3*a(n-3) - 3*a(n-4) + 3*a(n-5) + a(n-6) - a(n-7) for n>6.
(End)
E.g.f.: (x*(-3 + x + x^2)*cosh(x) + (3 - x + x^3)*sinh(x))/16. - Stefano Spezia, Mar 02 2020
EXAMPLE
For example, for n=6, a(6) = 9 and the 9 subsets are: {1,2,4}, {1,2,6}, {1,4,6}, {2,3,4}, {2,3,6}, {2,4,5}, {2,5,6}, {3,4,6}, {4,5,6}.
MATHEMATICA
a[n_] := Ceiling[n/2] * Binomial[Floor[n/2], 2]; Array[a, 55, 0] (* Amiram Eldar, Mar 01 2020 *)
Table[Length[Select[Subsets[Range[n], {3}], Total[Boole[OddQ[#]]]==1&]], {n, 0, 60}] (* Harvey P. Dale, Jul 26 2020 *)
PROG
(PARI) a(n) = ceil(n/2) * binomial(floor(n/2), 2) \\ Andrew Howroyd, Mar 01 2020
(PARI) concat([0, 0, 0, 0], Vec(x^4*(2 + x) / ((1 - x)^4*(1 + x)^3) + O(x^40))) \\ Colin Barker, Mar 02 2020
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Enrique Navarrete, Feb 29 2020
STATUS
approved