A097169 a(n) = Sum_{k=0..n} C(floor((n+1)/2),floor((k+1)/2)) * 3^k.

1, 4, 13, 52, 133, 604, 1333, 6772, 13333, 74284, 133333, 801892, 1333333, 8550364, 13333333, 90286612, 133333333, 945912844, 1333333333, 9846548932, 13333333333, 101952273724, 133333333333, 1050903796852, 1333333333333

Offset: 0

Author

Paul Barry, Jul 30 2004

Keywords

• easy
• nonn

Comments

a(n) = (4/3){1,10,10,100,100,1000...} -9{0,1,0,9,0,81...} -(1/3){1,1,1,1,1,1...} .

a(2n) = A097166(n).

a(2n+1)/4 = A097168(n).

Links

• Index entries for linear recurrences with constant coefficients, signature (1,19,-19,-90,90)

Crossrefs

Cf. A097162, A075427.

Programs

Mathematica

LinearRecurrence[{1,19,-19,-90,90},{1,4,13,52,133},30] (* Harvey P. Dale, Dec 15 2017 *)

Formula

G.f.: (1+3x-10x^2-18x^3)/((1-x)*(1-9x^2)*(1-10x^2)).

a(n) = 2((1-sqrt(10))(-sqrt(10))^n+(1+sqrt(10))(sqrt(10))^n)/3+3((-3)^n-3^n)/2-1/3.

a(n) = a(n-1) +19a(n-2) -19a(n-3) -90a(n-4) +90a(n-5).