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A129637 Number of n-step paths that can go {west, southeast, southwest, northwest} on a 240-degree wedge on the equilateral triangular lattice. 2
1, 3, 11, 41, 157, 607, 2367, 9277, 36505, 144059, 569779, 2257521, 8957109, 35579351, 141460391, 562871557, 2241129905, 8928207987, 35584894299, 141886838329, 565938926669 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

If we use the "hour hand" positions at 1, 3, 5, 7, 9, and 11 o'clock on a 12-hour clock to specify directions in the triangular lattice, the allowable steps are in directions 5, 7, 9, and 11 and the path is restricted to stay on or above the 1-7 line. In the Mathematica recurrence below, a(n,k) denotes the number of paths of length n ending k units from the 1-7 line, counted by the last step. - David Callan, Jul 22 2008

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..200

A. Bostan, Computer Algebra for Lattice Path Combinatorics, Seminaire de Combinatoire Ph. Flajolet, March 28 2013.

FORMULA

Recurrence: (-28-28*n)*a(n) + (-13-n)*a(1+n) + (21+6*n)*a(n+2) + (-4-n)*a(n+3), a(0) = 1, a(1) = 3, a(2) = 11.

G.f.: (1/4*i)*sqrt(-1+2*t+7*t^2)/((-1+4*t)*t)-(1/4)*(-1+5*t)/(t*(-1+4*t)), where i is the imaginary unit.

Differential equation: -(1+28*t^3-6*t+t^2)*t*((d/dt)f(t)) + (9*t-12*t^2-1-28*t^3)*f(t) + 1 - 3*t, f(0) = 1.

a(n) = Sum_{k=0..n+1}(binomial(n+1,k)*Sum_{i=0..n}(2^(i+k-1)*binomial(k,i)*(-1)^(n-i)*binomial(n-i-1,k-i-1)))/(n+1). - Vladimir Kruchinin, Feb 28 2016

a(n) ~ 2^(2*n-1). - Vaclav Kotesovec, Feb 28 2016

EXAMPLE

a(1) = 3 because only 3 out of the 4 steps are permissible from the origin;

a(2) = 11 because the northwest and west steps are followed by 4 permissible steps each, but the southwest step is only followed by 3 permissible steps.

MATHEMATICA

a[0, 0]=1; a[n_, k_]/; k<0 || k>n := 0; a[n_, k_]/; 0<=k<=n := a[n, k] = 2a[n-1, k-1] + a[n-1, k] + a[n-1, k+1]; a[n_]:=Sum[a[n, k], {k, 0, n}]; Table[a[n], {n, 0, 10}] (* David Callan, Jul 22 2008 *)

PROG

(Maxima)

a(n):=sum(binomial(n+1, k)*sum(2^(i+k-1)*binomial(k, i)*(-1)^(n-i)*binomial(n-i-1, k-i-1), i, 0, n), k, 0, n+1)/(n+1); /* Vladimir Kruchinin, Feb 28 2016 */

(PARI) a(n) = {sum(k=0, n+1, binomial(n+1, k) * sum(i=0, n, 2^(i+k-1)*binomial(k, i)*(-1)^(n-i)*binomial(n-i-1, k-i-1)))/(n+1)} \\ Andrew Howroyd, Dec 22 2017

CROSSREFS

Cf. A129400.

Sequence in context: A196472 A258471 A176085 * A084077 A027103 A151086

Adjacent sequences:  A129634 A129635 A129636 * A129638 A129639 A129640

KEYWORD

nonn

AUTHOR

Rebecca Xiaoxi Nie (rebecca.nie(AT)utoronto.ca), May 31 2007

STATUS

approved

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Last modified December 16 01:32 EST 2019. Contains 330013 sequences. (Running on oeis4.)