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A185828
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Half the number of n X 2 binary arrays with every element equal to exactly one or two of its horizontal and vertical neighbors.
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7
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1, 3, 10, 23, 61, 162, 421, 1103, 2890, 7563, 19801, 51842, 135721, 355323, 930250, 2435423, 6376021, 16692642, 43701901, 114413063, 299537290, 784198803, 2053059121, 5374978562, 14071876561, 36840651123, 96450076810, 252509579303
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OFFSET
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1,2
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COMMENTS
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LINKS
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FORMULA
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Empirical: a(n) = 2*a(n-1) + a(n-2) + 2*a(n-3) - a(n-4).
a(n) = n*Sum_{k=0..n-1} C(2n-2k, 2k)/(n-k). - Paul D. Hanna, Mar 19 2011
L.g.f.: Sum_{n>=1} a(n)*x^n/n = -log((1+x+x^2)*(1-3*x+x^2))/2. - Paul D. Hanna, Mar 19 2011
Logarithmic derivative of A051286, which is the Whitney number of level n of the lattice of the ideals of the fence of order 2n. - Paul D. Hanna, Mar 19 2011
Empirical g.f.: x*(1+x+3*x^2-2*x^3)/(1+x+x^2)/(1-3*x+x^2). - Colin Barker, Feb 22 2012
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EXAMPLE
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Some solutions for 4 X 2 with a(1,1)=0:
0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0
1 1 0 1 0 1 1 1 0 1 1 0 0 1 1 1 1 0 0 1
0 1 0 0 0 1 0 1 1 0 1 0 1 1 1 1 1 1 0 1
0 0 1 1 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1
The logarithmic g.f. begins:
L(x) = x + 3*x^2/2 + 10*x^3/3 + 23*x^4/4 + 61*x^5/5 + 162*x^6/6 + ..., where
exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 11*x^4 + 26*x^5 + 63*x^6 + ... + A051286(n)*x^n/n + ... - Paul D. Hanna, Mar 19 2011
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MAPLE
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a := proc(n): n*add(binomial(2*n-2*k, 2*k)/(n-k), k=0..n-1) end: seq(a(n), n=1..28); # Johannes W. Meijer, Jun 18 2018
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PROG
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(PARI) {a(n)=n*sum(k=0, n-1, binomial(2*n-2*k, 2*k)/(n-k))} /* Paul D. Hanna, Mar 19 2011 */
(PARI) {a(n)=n*polcoeff(-log( (1+x+x^2)*(1-3*x+x^2) +x*O(x^n))/2, n)} /* Paul D. Hanna, Mar 19 2011 */
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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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