

A178522


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<=n1).


7



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

0,4


COMMENTS

A Fibonacci tree of order n (n>=2) is a complete binary tree whose left subtree is the Fibonacci tree of order n1 and whose right subtree is the Fibonacci tree of order n2; 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=0..n1} k*T(n,k) = A178523(n).


REFERENCES

D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, AddisonWesley, Reading, MA, 1998, p. 417.


LINKS

Table of n, a(n) for n=0..86.
Y. Horibe, An entropy view of Fibonacci trees, Fibonacci Quarterly, 20, No. 2, 1982, 168178.


FORMULA

G.f.: G(t,z)=(1tz+tz^2)/[(1z)(1tztz^2)].
T(k,n) = T(k1,n1)+T(k1,n) with T(0,0)=1, T(k,0)=1 for k>0, T(0,n)=2 for n>0.  Frank M Jackson, Aug 30 2011


EXAMPLE

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,
...


MAPLE

G := (1t*z+t*z^2)/((1z)*(1t*zt*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 .. n1) end do; # yields sequence in triangular form


CROSSREFS

Cf. A001595, A059214, A178523, A067331, A002940. See A059250 for another version.
Sequence in context: A273673 A140186 A078498 * A131240 A263666 A107027
Adjacent sequences: A178519 A178520 A178521 * A178523 A178524 A178525


KEYWORD

nonn,tabf


AUTHOR

Emeric Deutsch, Jun 15 2010


STATUS

approved



