A330592 a(n) is the number of subsets of {1,2,...,n} that contain exactly two odd numbers.

0, 0, 2, 4, 12, 24, 48, 96, 160, 320, 480, 960, 1344, 2688, 3584, 7168, 9216, 18432, 23040, 46080, 56320, 112640, 135168, 270336, 319488, 638976, 745472, 1490944, 1720320, 3440640, 3932160, 7864320, 8912896, 17825792, 20054016, 40108032, 44826624, 89653248

Offset: 1

Author

Enrique Navarrete, Dec 18 2019

Keywords

• nonn
• easy

Comments

2*a(n-1) for n>1 is the number of subsets of {1,2,...,n} that contain exactly two even numbers. For example, for n=5, 2*a(4)=8 and the 8 subsets are {2,4}, {1,2,4}, {2,3,4}, {2,4,5}, {1,2,3,4}, {1,2,4,5}, {2,3,4,5}, {1,2,3,4,5}. - Enrique Navarrete, Dec 20 2019

Examples

			For n=5, a(5)=12 and the 12 subsets are {1,3}, {1,5}, {3,5}, {1,2,3}, {1,2,5}, {1,3,4}, {1,4,5}, {2,3,5}, {3,4,5}, {1,2,3,4}, {1,2,4,5}, {2,3,4,5}.
		

Links

• Colin Barker, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (0,6,0,-12,0,8).

Crossrefs

Cf. A089822 (with exactly two primes).

Programs

Magma

[IsEven(n) select Binomial(n div 2,2)*2^(n div 2) else Binomial((n+1) div 2,2)*2^((n-1) div 2):n in [1..40]]; // Marius A. Burtea, Dec 19 2019

Mathematica

a[n_] := If[OddQ[n], Binomial[(n + 1)/2, 2]*2^((n - 1)/2), Binomial[n/2, 2]*2^(n/2)]; Array[a, 38] (* Amiram Eldar, Mar 24 2022 *)

PARI

concat([0,0], Vec(2*x^3*(1 + 2*x) / (1 - 2*x^2)^3 + O(x^40))) \\ Colin Barker, Dec 20 2019

Formula

a(n) = binomial((n+1)/2,2) * 2^((n-1)/2), n odd;

a(n) = binomial(n/2,2) * 2^(n/2), n even.

G.f.: 2*(2*x+1)*x^3/(1-2*x^2)^3.

a(n) = 6*a(n-2) - 12*a(n-4) + 8*a(n-6) for n>6. - Colin Barker, Dec 20 2019

From Amiram Eldar, Mar 24 2022: (Start)

Sum_{n>=3} 1/a(n) = 3*(1-log(2)).

Sum_{n>=3} (-1)^(n+1)/a(n) = 1-log(2). (End)