A099099 Quadrisection of a generalized Padovan sequence.

1, 1, 1, 2, 6, 16, 37, 80, 172, 377, 839, 1874, 4175, 9274, 20577, 45665, 101393, 225193, 500162, 1110790, 2466760, 5477917, 12164896, 27015092, 59993817, 133231279, 295872778, 657057431, 1459155634, 3240410561, 7196122817

Offset: 0

Author

Paul Barry, Sep 29 2004

Keywords

• easy
• nonn

Comments

Quadrisection of sequence with g.f. 1/(1 - x^3 - x^4), or A017817.

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-1).

Programs

Mathematica

Join[{1},a=0;b=0;c=0;d=1;Table[a+=b;b+=c;c+=d;d+=a,{n,50}]] (* Vladimir Joseph Stephan Orlovsky, Nov 19 2010 *)

Formula

G.f.: (1-x)^3/((1-x)^4-x^3).

a(n) = sum_{k=0..2n} binomial(k, 4n-3k).

a(n) = 4a(n-1) - 6a(n-2) + 5a(n-3) - a(n-4).

a(n) = A017817(4n).

a(n) = sum_{k=0..floor((n+1)/2)} binomial(n+k, 4k). - Paul Barry, May 09 2005