A094790 Number of (s(0), s(1), ..., s(2n)) such that 0 < s(i) < 7 and |s(i) - s(i-1)| = 1 for i = 1,2,...,2*n, s(0) = 1, s(2n) = 3.

1, 3, 9, 28, 89, 286, 924, 2993, 9707, 31501, 102256, 331981, 1077870, 3499720, 11363361, 36896355, 119801329, 388991876, 1263047761, 4101088878, 13316149700, 43237262993, 140390505643, 455845099957, 1480119728920

Offset: 1

Author

Herbert Kociemba, Jun 11 2004

Keywords

• nonn
• easy

Comments

In general a(n) = (2/m)*Sum_{r=1..m-1} sin(r*j*Pi/m)*sin(r*k*Pi/m)*(2*cos(r*Pi/m))^(2n)) counts (s(0), s(1), ..., s(2n)) such that 0 < s(i) < m and |s(i) - s(i-1)| = 1 for i = 1,2,...,2n, s(0) = j, s(2n) = k.

With interpolated zeros (0,0,1,0,3,0,9,...), counts walks of length n between the first and third nodes of P_6. - Paul Barry, Jan 26 2005

Counts all paths of length (2*n+1), n >= 0, starting at the initial node and ending on the nodes 1, 2, 3, 4 and 5 on the path graph P_6, see the Maple program. - Johannes W. Meijer, May 29 2010

With offset 0 = the INVERT transform of A055588. - Gary W. Adamson, Apr 01 2011

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1956
• Nachum Dershowitz, Between Broadway and the Hudson: A Bijection of Corridor Paths, arXiv:2006.06516 [math.CO], 2020.
• László Németh and László Szalay, Sequences Involving Square Zig-Zag Shapes, J. Int. Seq., Vol. 24 (2021), Article 21.5.2.
• Index entries for linear recurrences with constant coefficients, signature (5,-6,1).

Crossrefs

Cf. A005021, A028495, A052975, A055588, A078038, A080937, A094789.

Programs

Magma

[n le 3 select 3^(n-1) else 5*Self(n-1) -6*Self(n-2) +Self(n-3): n in [1..31]]; // G. C. Greubel, Feb 12 2023

Maple

with(GraphTheory):G:=PathGraph(6): A:= AdjacencyMatrix(G): nmax:=24; n2:=2*nmax+1: for n from 0 to n2 do B(n):=A^n; a(n):=add(B(n)[k,1],k=1..5); od: seq(a(2*n+1),n=0..nmax); # Johannes W. Meijer, May 29 2010

Mathematica

f[n_]:= FullSimplify[ TrigToExp[(2/7)Sum[ Sin[Pi*k/7]Sin[3Pi*k/7](2Cos[Pi*k/7] )^(2n), {k,6}]]];
Table[f[n], {n, 25}] (* Robert G. Wilson v, Jun 18 2004 *)
LinearRecurrence[{5,-6,1},{1,3,9},30] (* Harvey P. Dale, Nov 19 2019 *)

PARI

Vec(x*(1-2*x)/(1-5*x+6*x^2-x^3)+O(x^99)) \\ Charles R Greathouse IV, Jun 14 2015

SageMath

@CachedFunction
def a(n): # a = A094790
    if (n<4): return 3^(n-1)
    else: return 5*a(n-1) - 6*a(n-2) + a(n-3)
[a(n) for n in range(1,41)] # G. C. Greubel, Feb 12 2023

Formula

a(n) = (2/7)*Sum_{k=1..6} sin(Pi*k/7)*sin(3*Pi*k/7)*(2*cos(Pi*k/7))^(2n).

a(n) = 5*a(n-1) - 6*a(n-2) + a(n-3).

G.f.: x*(1-2*x)/(1 - 5*x + 6*x^2 - x^3).

a(n) = rightmost term in M^n * [1,0,0] where M = the 3 X 3 matrix [2,1,1; 1,2,0; 1,0,1]. E.g., M^3 * [1,0,0] = [19,14,9]; right term = 9 = a(3). - Gary W. Adamson, Apr 04 2006