A007829 From random walks on complete directed triangle.

0, 0, 0, 0, 0, 6, 8, 28, 44, 100, 162, 318, 514, 942, 1518, 2672, 4302, 7380, 11882, 20040, 32276, 53810, 86710, 143396, 231204, 380152, 613286, 1004188, 1620864, 2645928, 4272744, 6959326, 11242518, 18281222, 29542078, 47978666, 77552928, 125836374, 203445784

Offset: 0

Author

Eric Bussian [ ebussian(AT)math.gatech.edu ]

Keywords

• nonn
• walk

Links

• Sean A. Irvine, Table of n, a(n) for n = 0..500
• E. Bussian, Email to N. J. A. Sloane, Oct. 1994
• Index entries for linear recurrences with constant coefficients, signature (2,3,-6,-4,5,5,-2,-2).

Crossrefs

Cf. A000931, A084338.

Programs

Maple

m:=35; S:=series(2*x^5*(3-2*x-3*x^2)/((1-x)*(1-x-x^2)*(1-2*x^2)*(1-x^2-x^3)), x, m+1): seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 11 2020

Mathematica

b[n_]:= b[n]= If[n==0, 1, If[n<3, 0, b[n-2] +b[n-3]]]; Table[2*(2 +Fibonacci[n+2] -2^Floor[n/2] -p[n+7] -p[n+5]), {n,0,35}] (* G. C. Greubel, Mar 11 2020 *)

Sage

def A007829_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 2*x^5*(3-2*x-3*x^2)/((1-x)*(1-x-x^2)*(1-2*x^2)*(1-x^2-x^3)) ).list()
A007829_list(35) # G. C. Greubel, Mar 11 2020

Formula

From Colin Barker, Feb 03 2018: (Start)

G.f.: 2*x^5*(3 - 2*x - 3*x^2) / ((1 - x)*(1 - x - x^2)*(1 - 2*x^2)*(1 - x^2 - x^3)).

a(n) = 2*a(n-1) + 3*a(n-2) - 6*a(n-3) - 4*a(n-4) + 5*a(n-5) + 5*a(n-6) - 2*a(n-7) - 2*a(n-8) for n>7.

(End)

From G. C. Greubel, Mar 11 2020: (Start)

a(n) = 2*(2 + Fibonacci(n+2) - 2^floor(n/2) - A084338(n+2)).

a(n) = 2*(2 + Fibonacci(n+2) - 2^floor(n/2) - b(n+7) - b(n+5)), where b(n) = A000931(n). (End)