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A129637
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Number of n-step paths that can go {west, south-east, south-west,north-west} on a 240 degree wedge on the equilateral triangular lattice.
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1
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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; internal format)
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OFFSET
| 0,2
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COMMENTS
| If we use the "hour hands" 1,3,5,7,9,11 on the 12 hour clock to specify directions in the triangular lattice, the allowable steps are in directions 5,7,9,11 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 (callan(AT)stat.wisc.edu), Jul 22 2008
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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
Generating function = (1/4*i)*sqrt(-1+2*t+7*t^2)/((-1+4*t)*t)-(1/4)*(-1+5*t)/(t*(-1+4*t))
Differential equation: -(1+28*t^3-6*t+t^2)*t*(diff(f(t), t))+(9*t-12*t^2-1-28*t^3)*f(t)+1-3*t, f(0) = 1
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EXAMPLE
| a(1) = 3 because only 3 out of the 4 steps are permissible from the origin;
a(2) = 11 because the north-west and west steps are followed by 4 permissible steps each, but the south-west step is only followed by 3 permissible steps.
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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 (callan(AT)stat.wisc.edu), Jul 22 2008
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CROSSREFS
| Cf. A129400.
Sequence in context: A076540 A196472 A176085 * A084077 A027103 A151086
Adjacent sequences: A129634 A129635 A129636 * A129638 A129639 A129640
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KEYWORD
| nonn
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AUTHOR
| Rebecca Xiaoxi Nie (rebecca.nie(AT)utoronto.ca), May 31 2007
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