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A205336
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Number of length n+1 nonnegative integer arrays starting and ending with 0 with adjacent elements unequal but differing by no more than 3.
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1
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0, 3, 6, 35, 138, 689, 3272, 16522, 83792, 434749, 2278888, 12093271, 64741330, 349470487, 1899418046, 10387322922, 57111322368, 315523027610, 1750681516380, 9751416039535, 54507046599094, 305650440453943, 1718956630038438
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OFFSET
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1,2
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COMMENTS
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Number of excursions (walks starting at the origin, ending on the x-axis, and never go below the x-axis in between) with n steps from {-3,-2,-1,1,2,3}. - David Nguyen, Dec 20 2016
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LINKS
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FORMULA
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a(n) = Sum_{i=1..n}((Sum_{l=0..i}(binomial(i,l)*(Sum_{j=0=(3*(i-l))/7}((-1)^j*binomial(i-l,j)*binomial(-l+3*(-l-2*j+i)-j+i-1,3*(-l-2*j+i)-j)))*(-1)^l))*a(n-i))/n, a(0)=1. - Vladimir Kruchinin, Apr 07 2017
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EXAMPLE
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Some solutions for n=5
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..3....3....3....1....3....1....1....3....3....2....1....3....1....3....3....3
..4....6....2....0....2....3....3....2....5....4....4....1....3....2....2....0
..2....5....5....3....4....4....2....3....4....1....2....2....0....4....0....2
..3....2....2....2....2....1....3....2....2....3....1....1....2....3....2....1
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
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MATHEMATICA
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a[n_] := a[n] = If[n == 0, 1, Sum[(Sum[Binomial[i, l] (Sum[(-1)^j Binomial[i - l, j] Binomial[-l + 3(-l - 2j + i) - j + i - 1, 3(-l - 2j + i) - j], {j, 0, (3(i - l))/7}]) (-1)^l, {l, 0, i}]) a[n - i], {i, 1, n}]/n];
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PROG
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(Maxima)
a(n):=if n=0 then 1 else sum((sum(binomial(i, l)*(sum((-1)^j*binomial(i-l, j)*binomial(-l+3*(-l-2*j+i)-j+i-1, 3*(-l-2*j+i)-j), j, 0, (3*(i-l))/7))*(-1)^l, l, 0, i))*a(n-i), i, 1, n)/n; /* Vladimir Kruchinin, Apr 07 2017 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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