A077937 Expansion of 1/(1-2*x-2*x^2+2*x^3).

1, 2, 6, 14, 36, 88, 220, 544, 1352, 3352, 8320, 20640, 51216, 127072, 315296, 782304, 1941056, 4816128, 11949760, 29649664, 73566592, 182532992, 452899840, 1123732480, 2788198656, 6918062592, 17165057536, 42589842944, 105673675776, 262196922368

Offset: 0

Author

N. J. A. Sloane, Nov 17 2002

Keywords

• nonn
• easy

Comments

Form the graph with matrix A = [1,1,1,1; 1,0,0,0; 1,0,0,0; 1,0,0,1]. Then the sequence 0, 1, 2, 6, ... counts walks of length n between the degree 5 vertex and the degree 3 vertex. - Paul Barry, Oct 02 2004

From Sean A. Irvine, Jun 05 2025: (Start)

Also, the number of walks of length n starting at vertex 0 in the graph:

1-2

/| |

0 | |

\| |

4-3. (End)

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Sean A. Irvine, Walks on Graphs.
• Index entries for linear recurrences with constant coefficients, signature (2,2,-2).

Crossrefs

Cf. A052528, A052987, A107300.

Programs

Magma

[n le 3 select Factorial(n) else 2*(Self(n-1) +Self(n-2) -Self(n-3)): n in [1..51]]; // G. C. Greubel, May 02 2022

Mathematica

LinearRecurrence[{2,2,-2}, {1,2,6}, 50] (* Vladimir Joseph Stephan Orlovsky, Jul 03 2011 *)
CoefficientList[Series[1/(1-2*x-2*x^2+2*x^3),{x,0,40}],x] (* Harvey P. Dale, Dec 05 2018 *)

PARI

Vec(1/(1-2*x-2*x^2+2*x^3)+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

SageMath

def A077937_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/(1-2*x-2*x^2+2*x^3) ).list()
A077937_list(50) # G. C. Greubel, May 02 2022

Formula

a(n) = 2*a(n-1) + 2*a(n-2) - 2*a(n-3) with a(0) = 1, a(1) = 2, and a(3) = 8. - G. C. Greubel, May 02 2022