

A129637


Number of nstep paths that can go {west, southeast, southwest, northwest} on a 240degree 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
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OFFSET

0,2


COMMENTS

If we use the "hour hand" positions at 1, 3, 5, 7, 9, and 11 o'clock on a 12hour 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 17 line. In the Mathematica recurrence below, a(n,k) denotes the number of paths of length n ending k units from the 17 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: (2828*n)*a(n) + (13n)*a(1+n) + (21+6*n)*a(n+2) + (4n)*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^36*t+t^2)*t*((d/dt)f(t)) + (9*t12*t^2128*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+k1)*binomial(k,i)*(1)^(ni)*binomial(ni1,ki1)))/(n+1).  Vladimir Kruchinin, Feb 28 2016
a(n) ~ 2^(2*n1).  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[n1, k1] + a[n1, k] + a[n1, 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+k1)*binomial(k, i)*(1)^(ni)*binomial(ni1, ki1), 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+k1)*binomial(k, i)*(1)^(ni)*binomial(ni1, ki1)))/(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



