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A293579
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Number of compositions of n where each part i is marked with a word of length i over a binary alphabet whose letters appear in alphabetical order and both letters occur at least once in the composition.
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2
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3, 16, 66, 248, 892, 3136, 10888, 37536, 128880, 441472, 1510176, 5161856, 17635264, 60233728, 205697152, 702386688, 2398283520, 8188622848, 27958448640, 95457597440, 325915589632, 1112751357952, 3799182641152, 12971244625920, 44286646775808, 151204164960256
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OFFSET
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2,1
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LINKS
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FORMULA
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a(n) = 6*a(n-1) - 10*a(n-2) + 4*a(n-3).
a(n) ~ 2^(n/2 - 2) * (1+sqrt(2))^(n+1).
a(n) = 2^(n/2 - 2) * ((sqrt(2)+1)^(n+1) - (sqrt(2)-1)^(n+1)) - 2^n.
(End)
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MAPLE
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b:= proc(n, k) option remember; `if`(n=0, 1,
add(b(n-j, k)*binomial(j+k-1, k-1), j=1..n))
end:
a:= n-> (k->add(b(n, k-i)*(-1)^i*binomial(k, i), i=0..k))(2):
seq(a(n), n=2..30);
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MATHEMATICA
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Table[Simplify[2^(n/2 - 2)*((Sqrt[2]+1)^(n+1) - (Sqrt[2]-1)^(n+1)) - 2^n], {n, 2, 20}] (* Vaclav Kotesovec, Oct 14 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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