A099098 Quadrisection of a Padovan sequence.

1, 1, 4, 12, 37, 114, 351, 1081, 3329, 10252, 31572, 97229, 299426, 922111, 2839729, 8745217, 26931732, 82938844, 255418101, 786584466, 2422362079, 7459895657, 22973462017, 70748973084, 217878227876, 670976837021, 2066337330754

Offset: 0

Author

Paul Barry, Sep 29 2004

Keywords

• easy
• nonn

Comments

Quadrisection of sequence with g.f. 1/(1-x^2-x^3), or A000931(n+3).

Examples

			1 + x + 4*x^2 + 12*x^3 + 37*x^4 + 114*x^5 + 351*x^6 + ...
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Sela Fried, Even-up words and their variants, arXiv:2505.14196 [math.CO], 2025. See p. 4.
• Index entries for linear recurrences with constant coefficients, signature (2,3,1).

Crossrefs

Bisection of A005251.

Programs

Mathematica

LinearRecurrence[{2,3,1},{1,1,4},40] (* Harvey P. Dale, Aug 23 2011 *)

Formula

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

a(n)=sum{k=0..2n, binomial(k, 4n-2k)};

a(n)=2a(n-1)+3a(n-2)+a(n-3);

a(n)=A000931(4n+3).

a(n) = Sum [k=0..n, C(2n-k, 2k) ].