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A151323
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Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, -1), (-1, 0), (0, -1), (0, 1), (1, 0), (1, 1)}.
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0
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1, 3, 14, 67, 342, 1790, 9580, 52035, 285990, 1586298, 8864676, 49844238, 281719164, 1599314652, 9113895960, 52109150691, 298806189318, 1717855010274, 9898828072692, 57158263594458, 330662400729492, 1916134078427556, 11120825740970088, 64634042348169294, 376139362185133404, 2191569966890629380
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OFFSET
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0,2
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LINKS
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FORMULA
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G.f. appears to be (((1+2*x)/(1-6*x))^(1/4)-1)/(2*x). [From Mark van Hoeij, Nov 20 2009]
Conjecture: (n+1)*a(n) -2*(2*n+1)*a(n-1) -12*(n-1)*a(n-2) = 0. - R. J. Mathar, Oct 26 2012
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MATHEMATICA
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aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, -1 + j, -1 + n] + aux[-1 + i, j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[i, 1 + j, -1 + n] + aux[1 + i, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[Sum[aux[i, j, n], {i, 0, n}, {j, 0, n}], {n, 0, 25}]
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CROSSREFS
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KEYWORD
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nonn,walk
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AUTHOR
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STATUS
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approved
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