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

0,3

Comments

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

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).

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) ].

Example

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

Mathematica

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

Crossrefs

Bisection of A005251.

Sequence in context: A101555 A033130 A196918 * A019481 A019480 A192907

Adjacent sequences: A099095 A099096 A099097 * A099099 A099100 A099101

Keyword

easy,nonn

Author

Paul Barry, Sep 29 2004

Status

approved