|
%I #36 Jul 07 2021 02:03:51
%S 0,1,1,3,4,10,15,34,55,117,199,406,714,1417,2548,4965,9061,17443,
%T 32148,61390,113887,216318,403051,762841,1425471,2691574,5039254,
%U 9500193,17809336,33539833,62928201,118428835,222324436,418214706,785402143,1476968554
%N Expansion of x*(1+x) / (1-3*x^2-x^3).
%C 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).
%H Michael De Vlieger, <a href="/A188022/b188022.txt">Table of n, a(n) for n = 0..3650</a>
%H Paul Barry, <a href="https://arxiv.org/abs/2104.01644">Centered polygon numbers, heptagons and nonagons, and the Robbins numbers</a>, arXiv:2104.01644 [math.CO], 2021.
%H L. E. Jeffery, <a href="/wiki/User:L._Edson_Jeffery/Unit-Primitive_Matrices">Unit-primitive matrices</a>
%H Kai Wang, <a href="https://www.researchgate.net/publication/337943524_Fibonacci_Numbers_And_Trigonometric_Functions_Outline">Fibonacci Numbers And Trigonometric Functions Outline</a>, (2019).
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,3,1).
%F a(n) = 3*a(n-2)+a(n-3).
%F a(n) = A187498(3*n+1).
%F a(n) = A052931(n-2)+A052931(n-1). - _R. J. Mathar_, Mar 22 2011
%t LinearRecurrence[{0, 3, 1}, {0, 1, 1}, 36] (* or *)
%t CoefficientList[Series[x (1 + x)/(1 - 3 x^2 - x^3), {x, 0, 35}], x] (* _Michael De Vlieger_, Mar 10 2020 *)
%Y Cf. A094832 (bisection), A094833 (bisection).
%Y Cf. A052931, A187498.
%K nonn,easy
%O 0,4
%A _L. Edson Jeffery_, Mar 18 2011
|
