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A033303
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Expansion of (1+x)/(1-2*x-x^2+x^3).
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3
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1, 3, 7, 16, 36, 81, 182, 409, 919, 2065, 4640, 10426, 23427, 52640, 118281, 265775, 597191, 1341876, 3015168, 6775021, 15223334, 34206521, 76861355, 172705897, 388066628, 871977798, 1959316327
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Also the number of the one-sided n-step prudent walks, avoiding 3 or more consecutive east steps. - Shanzhen Gao, April 27 2011
First differences are in A052534.
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REFERENCES
| S. Gao, H. Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks (submitted to INTEGERS: The Electronic Journal of Combinatorial Number Theory).
R. P. Stanley, Enumerative Combinatorics I, p. 244.
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FORMULA
| a(0)=1, a(1)=h(n), and a(n)=h(n)+h(n-1) for n>=2, where h(n) = sum(k=1..n, sum(j=0..k, binomial(k,j) * binomial(j,n-3*k+2*j) * 2^(3*k-n-j) * (-1)^(k-j))). [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), Sep 09 2010]
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PROG
| (Maxima) h(n):=sum(sum(binomial(k, j)*binomial(j, n-3*k+2*j)*2^(3*k-n-j)*(-1)^(k-j), j, 0, k), k, 1, n); a(n):=if n=0 then 1 else if n=2 then h(n) else h(n)+h(n-1); [From Kruchinin Vladimir (kru(AT)ie.tusur.ru), Sep 09 2010]
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CROSSREFS
| Sequence in context: A019489 A077852 A020746 * A078056 A173761 A124671
Adjacent sequences: A033300 A033301 A033302 * A033304 A033305 A033306
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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