A097162 a(n) = Sum_{k=0..n} C(floor((n+1)/2),floor((k+1)/2))*2^k.

1, 3, 7, 21, 37, 123, 187, 681, 937, 3663, 4687, 19341, 23437, 100803, 117187, 520401, 585937, 2667543, 2929687, 13599861, 14648437, 69047883, 73242187, 349433721, 366210937, 1763945823, 1831054687, 8886837981, 9155273437, 44702625363

Offset: 0

Author

Paul Barry, Jul 30 2004

Keywords

• easy
• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,9,-9,-20,20)

Crossrefs

Cf. A075427.

Cf. A057651, A097165.

Programs

Maple

A097162:=n->add(binomial(floor((n+1)/2),floor((k+1)/2))*2^k, k=0..n): seq(A097162(n), n=0..30); # Wesley Ivan Hurt, Sep 18 2014

Mathematica

LinearRecurrence[{1,9,-9,-20,20},{1,3,7,21,37},50] (* Vincenzo Librandi, Jan 30 2012 *)

Formula

G.f.: (1+2*x-5*x^2-4*x^3)/((1-x)*(1-4*x^2)*(1-5*x^2)).

a(n) = (3/4-3*sqrt(5)/4)*(-sqrt(5))^n +(3/4+3*sqrt(5)/4)*(sqrt(5))^n-(2^n-(-2)^n)-1/2.

a(2*n) = A057651(n); a(2*n+1)=3*A097165(n).

a(n+5) = 20*a(n)-20*a(n+1)-9*a(n+2)+9*a(n+3)+a(n+4). - Robert Israel, Sep 18 2014