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

1,3

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

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).

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)

Example

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}.

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 *)

Prog

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

Crossrefs

Cf. A089822 (with exactly two primes).

Sequence in context: A230481 A212169 A108720 * A089822 A079352 A089888

Adjacent sequences: A330589 A330590 A330591 * A330593 A330594 A330595

Keyword

nonn,easy

Author

Enrique Navarrete, Dec 18 2019

Status

approved