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A081038
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3rd binomial transform of (1,2,0,0,0,0,0,0,...).
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23
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1, 5, 21, 81, 297, 1053, 3645, 12393, 41553, 137781, 452709, 1476225, 4782969, 15411789, 49424013, 157837977, 502211745, 1592728677, 5036466357, 15884240049, 49977243081, 156905298045, 491636600541, 1537671920841
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OFFSET
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0,2
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COMMENTS
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a(n) is the number of distinguished parts in all compositions of n+1 in which some (possibly all or none) of the parts have been distinguished. a(1) = 2 because we have: 2', 1'+1, 1+1', 1'+1' where we see 5's marking the distinguished parts. With offset=1, a(n) = Sum_{k=1..n} A200139(n,k)*k. - Geoffrey Critzer, Jan 12 2013
For n>=1, a(n-1) the number of ternary strings of length 2n containing the block 11..12 with n ones where no runs of length larger than n are permitted. - Marko Riedel, Mar 08 2016
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LINKS
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FORMULA
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G.f.: (1-x)/(1-3*x)^2.
a(n) = 6*a(n-1) - 9*a(n-2), with a(0)=1, a(1)=5.
a(n) = (2*n+3)*3^(n-1).
a(n) = Sum_{k=0..n} (k+1)*2^k*binomial(n, k).
a(n) = 2*A086972(n) - 1. - Lambert Herrgesell (zero815(AT)googlemail.com), Feb 10 2008
Sum_{n>=0} 1/a(n) = 9*(sqrt(3)*arctanh(1/sqrt(3)) - 1).
Sum_{n>=0} (-1)^n/a(n) = 9 - 3*sqrt(3)*Pi/2. (End)
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MAPLE
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MATHEMATICA
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LinearRecurrence[{6, -9}, {1, 5}, 40] (* Harvey P. Dale, Jun 22 2012 *)
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PROG
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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