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A178522
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Triangle read by rows: T(n,k) is the number of nodes at level k in the Fibonacci tree of order n (n>=0, 0<=k<=n-1).
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7
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1, 1, 1, 2, 1, 2, 2, 1, 2, 4, 2, 1, 2, 4, 6, 2, 1, 2, 4, 8, 8, 2, 1, 2, 4, 8, 14, 10, 2, 1, 2, 4, 8, 16, 22, 12, 2, 1, 2, 4, 8, 16, 30, 32, 14, 2, 1, 2, 4, 8, 16, 32, 52, 44, 16, 2, 1, 2, 4, 8, 16, 32, 62, 84, 58, 18, 2, 1, 2, 4, 8, 16, 32, 64, 114, 128, 74, 20, 2, 1, 2, 4, 8, 16, 32, 64, 126
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OFFSET
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0,4
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COMMENTS
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A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n-1 and whose right subtree is the Fibonacci tree of order n-2; each of the Fibonacci trees of order 0 and 1 is defined as a single node.
Sum of entries in row n is A001595(n).
Sum(k*T(n,k), k>=0)=A178523.
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REFERENCES
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Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.
D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.
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LINKS
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Table of n, a(n) for n=0..86.
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FORMULA
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G.f.=G(t,z)=(1-tz+tz^2)/[(1-z)(1-tz-tz^2)].
Comment from Frank M Jackson, Aug 30 2011: T(k,n)=T(k-1,n-1)+T(k-1,n) with T(0,0)=1, T(k,0)=1 for k>0, T(0,n)=2 for n>0.
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EXAMPLE
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Triangle starts:
1,
1,
1,2,
1,2,2,
1,2,4,2,
1,2,4,6,2,
1,2,4,8,8,2,
1,2,4,8,14,10,2,
1,2,4,8,16,22,12,2,
1,2,4,8,16,30,32,14,2,
...
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MAPLE
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G := (1-t*z+t*z^2)/((1-z)*(1-t*z-t*z^2)): Gser := simplify(series(G, z = 0, 17)): for n from 0 to 15 do P[n] := sort(coeff(Gser, z, n)) end do: 1; for n to 13 do seq(coeff(P[n], t, k), k = 0 .. n-1) end do; # yields sequence in triangular form
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CROSSREFS
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Cf. A001595, A059214, A178523, A067331, A002940. See A059250 for another version.
Sequence in context: A132749 A140186 A078498 * A131240 A107027 A107030
Adjacent sequences: A178519 A178520 A178521 * A178523 A178524 A178525
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch, Jun 15 2010
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STATUS
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approved
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