A002707 Number of ternary trees with n nodes. (Formerly M3128 N1268)

1, 1, 3, 31, 8401, 100130704103, 167321084032306100365838018661441, 780729803567659232170209385771412393899822868427528396719062233553621064241305757906662806984051

Offset

0,3

Comments

Consider the free algebraic system with one ternary commutative b(x,y,z) operator and one generator A. The number of elements of height n is a(n) where the height of A is zero and the height of b(x,y,z) is one more than the maximum heights of x, y and z. - Michael Somos, Mar 06 2012

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Table of n, a(n) for n=0..7.

Z. A. Melzak, A note on homogeneous dendrites, Canad. Math. Bull., 11 (1968), 85-93.

Index entries for sequences related to trees

Formula

a(n) = a(n-1)*(a(n-1)+a(n-2))*(a(n-1)+a(n-2)+a(n-3))/6 + a(n-1)*(a(n-1)+a(n-2))/(a(n-2)+a(n-3))*(a(n-1)/a(n-2)-a(n-2)/a(n-3)) + a(n-1)^2/a(n-2). - James A. Sellers, Feb 14 2000

Mathematica

RecurrenceTable[{a[0]==a[1]==1, a[2]==3, a[n]==a[n-1](a[n-1]+a[n-2])(a[n-1]+a[n-2]+a[n-3])/6+ a[n-1](a[n-1]+a[n-2])/(a[n-2]+a[n-3])(a[n-1]/a[n-2]-a[n-2]/a[n-3])+a[n-1]^2/a[n-2]}, a, {n, 10}] (* Harvey P. Dale, Feb 27 2012 *)
SetAttributes[b, Orderless]; S[0] = {A}; S[n_] := S[n] = Union @ Flatten[ Table[ Outer[ b, S[k1], S[k2], S[n - 1]], {k1, 0, n - 1}, {k2, 0, n - 1}], 4]; a[ n_] := Length @ S[n]; (* Michael Somos, Mar 06 2012 *)

Crossrefs

Sequence in context: A022499 A129209 A134721 * A283247 A261472 A144964

Adjacent sequences: A002704 A002705 A002706 * A002708 A002709 A002710

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, Feb 14 2000

Status

approved