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A131889
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a(n) is the number of shapes of balanced trees with constant branching factor 3 and n nodes. The node is balanced if the size, measured in nodes, of each pair of its children differ by at most one node.
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6
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1, 1, 3, 3, 1, 9, 27, 27, 81, 81, 27, 27, 9, 1, 27, 243, 729, 6561, 19683, 19683, 59049, 59049, 19683, 177147, 531441, 531441, 1594323, 1594323, 531441, 531441, 177147, 19683, 59049, 59049, 19683, 19683, 6561, 729, 243, 27, 1, 81, 2187, 19683, 531441, 4782969
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OFFSET
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0,3
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COMMENTS
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a(n) is always an integer power of 3.
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LINKS
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FORMULA
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a(0) = a(1) = 1; a(3n+1+m) = (3 choose m) * a(n+1)^m * a(n)^(3-m), where n >= 0 and 0 <= m <= 3.
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MAPLE
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a:= proc(n) option remember; local m, r; if n<2 then 1 else
r:= iquo(n-1, 3, 'm'); binomial(3, m) *a(r+1)^m *a(r)^(3-m) fi
end:
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MATHEMATICA
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a[n_, k_] := a[n, k] = Module[{m, r}, If[n < 2 || k == 1, 1, If[k == 0, 0, {r, m} = QuotientRemainder[n - 1, k]; Binomial[k, m]*a[r + 1, k]^m*a[r, k]^(k - m)]]];
a[n_] := a[n, 3];
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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