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A190277 Number of trails between opposite vertices in a triangle strip. 0
1, 1, 2, 4, 9, 23, 62, 174, 497, 1433, 4150, 12044, 34989, 101695, 295642, 859566, 2499277, 7267081, 21130538, 61441732, 178655937, 519483767, 1510520966, 4392195390, 12771343961, 37135696841 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

a(n) is the number of trails from 1 to n in an undirected graph with vertex set {1, 2, ..., n}, where i and j are adjacent if and only if |i-j|=1 or |i-j|=2. A trail can visit the same vertex more than once, but it cannot repeat an edge.

LINKS

Table of n, a(n) for n=1..26.

StackExchange, Counting trails in a triangular grid

Index entries for linear recurrences with constant coefficients, signature (3, 1, -3, -2).

FORMULA

a(n) = 3*a(n-1) + a(n-2) - 3*a(n-3) - 2*a(n-4) for n > 4.

G.f.: x*(1-2*x-2*x^2)/(1-3x-x^2+3*x^3+2*x^4).

EXAMPLE

For n = 5 there are 9 trails: 12345, 1235, 12435, 1245, 132435, 13245, 134235, 1345, and 135.

MAPLE

a := [1, 1, 2, 4, seq(0, i = 1 .. 36)]: for n from 5 to 40 do a[n] := 3*a[n-1]+a[n-2]-3*a[n-3]-2*a[n-4] end do; a;

MATHEMATICA

a[1] = 1; a[2] = 1; a[3] = 2; a[4] = 4; a[n_] := a[n] = 3 a[n - 1] + a[n - 2] - 3 a[n - 3] - 2 a[n - 4]; Table[a[n], {n, 1, 40}]

LinearRecurrence[{3, 1, -3, -2}, {1, 1, 2, 4}, 30] (* Harvey P. Dale, May 24 2011 *)

CROSSREFS

Sequence in context: A213683 A032010 A032028 * A274882 A127384 A058585

Adjacent sequences:  A190274 A190275 A190276 * A190278 A190279 A190280

KEYWORD

nonn,easy,walk

AUTHOR

David Radcliffe, May 07 2011

STATUS

approved

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Last modified May 26 22:57 EDT 2019. Contains 323597 sequences. (Running on oeis4.)