A178522 - OEIS

%I #20 Jan 05 2025 19:51:39

%S 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,

%T 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,

%U 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

%N 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).

%C 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.

%C Sum of entries in row n is A001595(n).

%C Sum_{k=0..n-1} k*T(n,k) = A178523(n).

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

%H Y. Horibe, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/20-2/horibe.pdf">An entropy view of Fibonacci trees</a>, Fibonacci Quarterly, 20, No. 2, 1982, 168-178.

%F G.f.: G(t,z)=(1-tz+tz^2)/[(1-z)(1-tz-tz^2)].

%F 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. - _Frank M Jackson_, Aug 30 2011

%e Triangle starts:

%e 1,

%e 1,

%e 1,2,

%e 1,2,2,

%e 1,2,4,2,

%e 1,2,4,6,2,

%e 1,2,4,8,8,2,

%e 1,2,4,8,14,10,2,

%e 1,2,4,8,16,22,12,2,

%e 1,2,4,8,16,30,32,14,2,

%e ...

%p 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

%Y Cf. A001595, A059214, A178523, A067331, A002940. See A059250 for another version.

%K nonn,tabf

%O 0,4

%A _Emeric Deutsch_, Jun 15 2010