A094556 -id:A094556 - cp's OEIS frontend

A094555 Number of walks of length n between two vertices on the same triangular face of a truncated tetrahedron (triangular prism).

0, 1, 1, 6, 11, 46, 111, 386, 1051, 3366, 9671, 29866, 87891, 267086, 794431, 2396946, 7163531, 21545206, 64526391, 193797626, 580955971, 1743741726, 5229477551, 15691927906, 47068793211, 141220360646, 423633119911, 1270955283786

Offset: 0

Paul Barry, May 11 2004

Average of binomial and inverse binomial transforms of the Jacobsthal numbers A001045. - Paul Barry, Jan 04 2005

Andrew Howroyd, Table of n, a(n) for n = 0..1000
R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 3.
Index entries for linear recurrences with constant coefficients, signature (2,5,-6).

Cf. A001045, A094554, A094556.

LinearRecurrence[{2, 5, -6}, {0, 1, 1, 6}, 30] (* Greg Dresden, Jun 19 2021 *)

a(n) = if(n==0, 0, (3^n - (-2)^n + 1)/6) \\ Andrew Howroyd, Jun 15 2021

G.f.: x*(1 - x - x^2)/((1 - x)*(1 + 2*x)*(1 - 3*x)).
a(n) = 3^n/6 - (-2)^n/6 + 1/6 - 0^n/6.
a(n) = 2*a(n-1) + 5*a(n-2) - 6*a(n-3) for n >= 4.
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k)*A001045(n-2k). - Paul Barry, Jan 04 2005
E.g.f.: exp(-2*x)*(exp(5*x) + exp(3*x) - exp(2*x) - 1)/6. - Stefano Spezia, Dec 26 2021

A094554 Number of closed walks of length n at a base vertex of a truncated tetrahedron (triangular prism).

Original entry on oeis.org

1, 0, 3, 2, 19, 30, 143, 322, 1179, 3110, 10183, 28842, 89939, 262990, 802623, 2380562, 7196299, 21479670, 64657463, 193535482, 581480259, 1742693150, 5231574703, 15687733602, 47077181819, 141203583430, 423666674343

Offset: 0

Paul Barry, May 11 2004

For n > 0, 6*a(n) is the number of 3-colorings of the prism of size 2 X n (i.e., C_2 X C_n).More generally, the number of k-colorings of the prism of size 2 X n is given by (k^2 - 3*k + 3)^n + (k - 1) * ((3 - k)^n + (1 - k)^n) + k^2 - 3*k + 1 (chromatic polynomial). - Sela Fried, Oct 07 2023

Andrew Howroyd, Table of n, a(n) for n = 0..1000
N. L. Biggs, R. M. Damerell, and D. A. Sands, Recursive families of graphs, Journal of Combinatorial Theory Series B Volume 12 (1972), 123-131.
R. J. Mathar, Counting Walks on Finite Graphs, Nov 2020, Section 3.
Index entries for linear recurrences with constant coefficients, signature (2,5,-6).

Cf. A002001, A025192, A094555, A094556, A328778.

LinearRecurrence[{2, 5, -6}, {1, 0, 3, 2}, 30] (* Greg Dresden, Jun 19 2021 *)

a(n) = if(n==0, 1, (1 + 3^n + 2*(-2)^n)/6) \\ Andrew Howroyd, Jun 14 2021

G.f.: (1 - 2*x - 2*x^2 + 2*x^3)/((1 - x)*(1 + 2*x)*(1 - 3*x)).
a(n) = 1/6 + 3^n/6 + (-2)^n/3 for n > 0.
a(n) = 2*a(n-1) + 5*a(n-2) - 6*a(n-3) for n >= 4.
E.g.f.: exp(-2*x)*(1 + exp(2*x))*(2 + exp(3*x))/6. - Stefano Spezia, Sep 26 2023

A344747 a(n) = (1/6)*(3^n + (-2)^n - 1).

Original entry on oeis.org

0, 2, 3, 16, 35, 132, 343, 1136, 3195, 10012, 29183, 89256, 264355, 799892, 2386023, 7185376, 21501515, 64613772, 193622863, 581305496, 1743042675, 5230875652, 15689131703, 47074385616, 141209175835, 423655489532, 1270910544543, 3812843481736, 11438306748995

Offset: 1

Ryan Brooks, Jun 14 2021

			a(4) = (1/6)*(3^4 + (-2)^4 - 1) = (1/6)*(81+16-1) = 16.

Index entries for linear recurrences with constant coefficients, signature (2,5,-6).

Potentially related to A094554, A094555, and A094556 (which have the same recurrence).

LinearRecurrence[{2, 5, -6}, {0, 2, 3}, 30] (* Greg Dresden, Jun 19 2021 *)

G.f.: x*(2 - x)/((1 - x)*(1 + 2*x)*(1 - 3*x)). - Andrew Howroyd, Jun 15 2021

a(n) = A094554(n-1) + 2*A094556(n-1). - Greg Dresden, Jun 19 2021

cp's OEIS Frontend

A094555 Number of walks of length n between two vertices on the same triangular face of a truncated tetrahedron (triangular prism).

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A094554 Number of closed walks of length n at a base vertex of a truncated tetrahedron (triangular prism).

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A344747 a(n) = (1/6)*(3^n + (-2)^n - 1).

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