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A129400
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Number of walks of length n on one 60-degree wedge of the equilateral triangular lattice. The walk can go along the walls of the wedge, but cannot cross the walls.
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2
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1, 2, 8, 32, 144, 672, 3264, 16256, 82688, 427520, 2240512, 11874304, 63533056, 342712320, 1861779456, 10176823296, 55932813312, 308907737088, 1713473323008, 9541666209792, 53322206674944, 298943898451968
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Counts colored Motzkin paths where each of the steps has two possible colors. Series reversion of x/(1+2x+4x^2). - Paul Barry (pbarry(AT)wit.ie), Sep 04 2007
Hankel transform is 4^C(n+1,2). [From Paul Barry (pbarry(AT)wit.ie), Oct 01 2009]
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FORMULA
| a(n)=2^n*A001006(n)=sum{k=0..floor(n/2), C(n,2k)*C(k)2^(n-2k)*2^k*2^k} where C(n)=A000108(n). - Paul Barry (pbarry(AT)wit.ie), Sep 04 2007
G.f.: 1/(1-2x-4x^2/(1-2x-4x^2/(1-2x-4x^2/(1-2x-4x^2/(1-.... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Oct 01 2009]
G.f.: (1/(8*x^2)) * (1-2*x-(1-4*x-12*x^2)^(1/2)) [From Mark van Hoeij (hoeij(AT)math.fsu.edu), Nov 02 2009]
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EXAMPLE
| a(1) = 2 because we can go east or north-east.
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MAPLE
| countwalk2 := proc (i::integer, j::integer, n::integer) option remember: if n < 0 or j < 0 or i < j then 0 elif n = 0 and i = 0 and j = 0 then 1 elif n = 0 then 0 else procname(i-2, j, n-1)+procname(i+2, j, n-1)+procname(i-1, j+1, n-1)+procname(i+1, j+1, n-1)+procname(i+1, j-1, n-1)+procname(i-1, j-1, n-1) end if end proc: counter2 := proc (n::nonnegint) option remember: add(add(countwalk2(i, j, n), i = 0 .. 2*n), j = 0 .. n) end proc:
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CROSSREFS
| Sequence in context: A150862 A150863 A084137 * A003304 A150864 A202814
Adjacent sequences: A129397 A129398 A129399 * A129401 A129402 A129403
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KEYWORD
| nonn
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AUTHOR
| Rebecca Xiaoxi Nie (rebecca.nie(AT)utoronto.ca), May 28 2007
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