A188022 Expansion of x*(1+x) / (1-3*x^2-x^3).

0, 1, 1, 3, 4, 10, 15, 34, 55, 117, 199, 406, 714, 1417, 2548, 4965, 9061, 17443, 32148, 61390, 113887, 216318, 403051, 762841, 1425471, 2691574, 5039254, 9500193, 17809336, 33539833, 62928201, 118428835, 222324436, 418214706, 785402143, 1476968554

Offset: 0

Author

L. Edson Jeffery, Mar 18 2011

Keywords

• nonn
• easy

Comments

Define the 4 X 4 tridiagonal unit-primitive matrix (see [Jeffery]) M=A_{9,1}=[0,1,0,0; 1,0,1,0; 0,1,0,1; 0,0,1,1]; then a(n)=[M^n](3,4)=[M^n](4,3).

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..3650
• Paul Barry, Centered polygon numbers, heptagons and nonagons, and the Robbins numbers, arXiv:2104.01644 [math.CO], 2021.
• L. E. Jeffery, Unit-primitive matrices
• Kai Wang, Fibonacci Numbers And Trigonometric Functions Outline, (2019).
• Index entries for linear recurrences with constant coefficients, signature (0,3,1).

Crossrefs

Cf. A094832 (bisection), A094833 (bisection).

Cf. A052931, A187498.

Programs

Mathematica

LinearRecurrence[{0, 3, 1}, {0, 1, 1}, 36] (* or *)
CoefficientList[Series[x (1 + x)/(1 - 3 x^2 - x^3), {x, 0, 35}], x] (* Michael De Vlieger, Mar 10 2020 *)

Formula

a(n) = 3*a(n-2)+a(n-3).

a(n) = A187498(3*n+1).

a(n) = A052931(n-2)+A052931(n-1). - R. J. Mathar, Mar 22 2011