A085810 Number of three-choice paths along a corridor of height 5, starting from the lower side.

1, 2, 5, 13, 35, 96, 266, 741, 2070, 5791, 16213, 45409, 127206, 356384, 998509, 2797678, 7838801, 21963661, 61540563, 172432468, 483144522, 1353740121, 3793094450, 10628012915, 29779028189, 83438979561, 233790820762, 655067316176, 1835457822857, 5142838522138, 14409913303805

Offset: 1

Author

Philippe Deléham, Jul 25 2003

Keywords

• easy
• nonn

Comments

From Svjetlan Feretic, Jun 01 2013: (Start)

A three-choice path is a path whose steps lie in the set {(1,1), (1,0), (1,-1)}.

The paths under consideration "live" in a corridor like 0<=y<=5. Thus, the ordinate of a vertex of a path can take six values (0,1,2,3,4,5), but the height of the corridor is five.

a(1)=1 is the number of paths with zero steps, a(2)=2 is the number of paths with one step, a(3)=5 is the number of paths with two steps, ...

Narrower corridors produce A000012, A000079, A000129, A001519, A057960. An infinitely wide corridor would produce A005773.

(End)

Diagonal sums of A114164. - Paul Barry, Nov 15 2005

C(n):= a(n)*(-1)^n appears in the following formula for the nonpositive powers of rho*sigma, where rho:=2*cos(Pi/7) and sigma:=sin(3*Pi/7)/sin(Pi/7) = rho^2-1 are the ratios of the smaller and larger diagonal length to the side length in a regular 7-gon (heptagon). See the Steinbach reference where the basis <1,rho,sigma> is used in an extension of the rational field. (rho*sigma)^(-n) = C(n) + B(n)*rho + A(n)*sigma,n>=0, with B(n)= A181880(n-2)*(-1)^n, and A(n)= A116423(n+1)*(-1)^(n+1). For the nonnegative powers see A120757(n), |A122600(n-1)| and A181879(n), respectively. See also a comment under A052547.

a(n) is also the number of bi-wall directed polygons with n cells. (The definition of bi-wall directed polygons is given in the article on A122737.)

Links

• G. C. Greubel, Table of n, a(n) for n = 1..1000
• Svjetlan Feretic, Generating functions for bi-wall directed polygons, in: Proc. of the Seventh Int. Conf. on Lattice Path Combinatorics and Applications (eds. S. Rinaldi and S. G. Mohanty), Siena, 2010, 147-151.
• Peter Steinbach, Golden fields: a case for the heptagon, Math. Mag. 70 (1997), p. 22-31 (formula 5).
• Roman Witula, Damian Slota and Adam Warzynski, Quasi-Fibonacci Numbers of the Seventh Order, J. Integer Seq., 9 (2006), Article 06.4.3.
• Index entries for linear recurrences with constant coefficients, signature (4,-3,-1).

Crossrefs

Cf. A000012, A000079, A000129, A001519, A057960, A005773.

Programs

Magma

I:=[1,2,5]; [n le 3 select I[n] else 4*Self(n-1)-3*Self(n-2)-Self(n-3): n in [1..35]]; // Vincenzo Librandi, Sep 18 2015

Mathematica

LinearRecurrence[{4,-3,-1}, {1,2,5}, 50] (* Roman Witula, Aug 09 2012 *)
CoefficientList[Series[(1 - 2 x)/(1 - 4 x + 3 x^2 + x^3), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 18 2015 *)

PARI

x='x+O('x^30); Vec((1-2*x)/(1-4*x+3*x^2+x^3)) \\ G. C. Greubel, Apr 19 2018

Formula

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

From Paul Barry, Nov 15 2005: (Start)

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

a(n) = Sum_{k=0..floor(n/2)} (Sum_{j=0..n-k} C(n-k, j)*C(j+k, 2k));

a(n) = Sum_{k=0..floor(n/2)} (Sum_{j=0..n-k} C(n-k, k+j)*C(k, k-j)*2^(n-2k-j));

a(n) = Sum_{k=0..floor(n/2)} (Sum_{j=0..n-2*k} C(n-j, n-2*k-j)*C(k, j)(-1)^j*2^(n-2*k-j)). (End)

a(n-1) = -B(n;-1) = (1/7)*((c(4)-c(1))*(1-c(1))^n + (c(1)-c(2))*(1-c(2))^n + (c(2)-c(4))*(1-c(4))^n), where a(-1):=0, c(j):=2*cos(2*Pi*j/7). Moreover, B(n;d), n in N, d in C, denotes the respective quasi-Fibonacci number defined in comments to A121449 or in Witula-Slota-Warzynski's paper (see also A077998, A006054, A052975, A094789, A121442). - Roman Witula, Aug 09 2012

Extensions

Name corrected and clarified, and offset 1 from Svjetlan Feretic, Jun 01 2013