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A178524 Triangle read by rows: T(n,k) is the number of leaves at level k in the Fibonacci tree of order n (n>=0, 0<=k<=n-1).

%I A178524 #20 Jan 05 2025 19:51:39
%S A178524 1,1,0,2,0,1,2,0,0,3,2,0,0,1,5,2,0,0,0,4,7,2,0,0,0,1,9,9,2,0,0,0,0,5,
%T A178524 16,11,2,0,0,0,0,1,14,25,13,2,0,0,0,0,0,6,30,36,15,2,0,0,0,0,0,1,20,
%U A178524 55,49,17,2,0,0,0,0,0,0,7,50,91,64,19,2,0,0,0,0,0,0,1,27,105,140,81,21,2
%N A178524 Triangle read by rows: T(n,k) is the number of leaves at level k in the Fibonacci tree of order n (n>=0, 0<=k<=n-1).
%C A178524 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 A178524 Sum of entries in row n is the Fibonacci number F(n+1) (A000045(n+1)).
%C A178524 Sum(k*T(n,k),k>=0)=A067331(n-2).
%C A178524 T(n,k) is the number of vertices in the Fibonacci cube G_{n-1} that have eccentricity k (see Klavzar and Mollard reference). - _Michel Mollard_, Aug 20 2014
%D A178524 D. E. Knuth, The Art of Computer Programming, Vol. 3, 2nd edition, Addison-Wesley, Reading, MA, 1998, p. 417.
%H A178524 A. Castro and M. Mollard, <a href="http://dx.doi.org/10.1016/j.disc.2011.11.006">The eccentricity sequences of Fibonacci and Lucas cubes</a>, Discrete Math., 312 (2012), 1025-1037. See Table 1. [From _N. J. A. Sloane_, Mar 22 2012]
%H A178524 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.
%H A178524 S. Klavzar, M. Mollard, <a href="http://dx.doi.org/10.1007/s00026-014-0233-x">Asymptotic Properties of Fibonacci Cubes and Lucas Cubes</a>, Annals of Combinatorics, 18, 2014, 447-457.
%F A178524 G.f.: G(t,z) = (1+z-t*z) / (1-t*z-t*z^2).
%e A178524 Triangle starts:
%e A178524 1,
%e A178524 1,
%e A178524 0, 2,
%e A178524 0, 1, 2,
%e A178524 0, 0, 3, 2,
%e A178524 0, 0, 1, 5, 2,
%e A178524 0, 0, 0, 4, 7, 2,
%e A178524 0, 0, 0, 1, 9, 9, 2,
%e A178524 0, 0, 0, 0, 5, 16, 11, 2,
%e A178524 0, 0, 0, 0, 1, 14, 25, 13, 2,
%e A178524 0, 0, 0, 0, 0, 6, 30, 36, 15, 2,
%p A178524 G := (1+z-t*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 A178524 Cf. A000045, A067331, A178522, A004070
%K A178524 nonn,tabf
%O A178524 0,4
%A A178524 _Emeric Deutsch_, Jun 15 2010