A334293 First quadrisection of Padovan sequence.

1, 0, 2, 5, 16, 49, 151, 465, 1432, 4410, 13581, 41824, 128801, 396655, 1221537, 3761840, 11584946, 35676949, 109870576, 338356945, 1042002567, 3208946545, 9882257736, 30433357674, 93722435101, 288627200960, 888855064897, 2737314167775, 8429820731201, 25960439030624

Offset: 0

Author

Oboifeng Dira, Apr 21 2020

Keywords

• nonn
• easy

Examples

			For n=3, a(3) = 2*a(2) + 3*a(1) + a(0) = 2*2 + 3*0 + 1 = 5.
		

Links

• Colin Barker, 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

Quadrisection of A000931.

Bisection (even part) of A099529.

Cf. A099098, A091140.

Programs

PARI

Vec((1 - 2*x - x^2) / (1 - 2*x - 3*x^2 - x^3) + O(x^30)) \\ Colin Barker, Apr 27 2020

Formula

a(n) = A000931(4n).

a(n) = A099529(2n).

a(n) = Sum_{k=0..n} binomial(2*n-k-1, 2*k-1).

a(n) = 2*a(n-1)+3*a(n-2)+a(n-3), a(0)=1, a(1)=0, a(2)=2 for n>=3.

G.f.: (1 - 2*x - x^2) / (1 - 2*x - 3*x^2 - x^3). - Colin Barker, Apr 27 2020