0,2

The n-th term of the coordination sequence of the infinite tree with valency 2m is the same as the number of reduced words of size n in the free group on m generators. In the five sequences A003946, A003948, A003950, A003952, A003954 m is 2, 3, 4, 5, 6. - Avi Peretz (njk(AT)netvision.net.il), Feb 23 2001 and Ola Veshta (olaveshta(AT)my-deja.com), Mar 30 2001.

a(n) is the number of nonreversing random walks of the length of n edges on a two-dimensional square lattice, all beginning at a fixed point P. - Pawel P. Mazur (Pawel.Mazur(AT)pwr.wroc.pl), Apr 06 2005

Binomial transform of {1, 3, 5, 11, 21, 43, ...}, see A001045 . Binomial transform is {1, 5, 21, 85, 341, 1365, ...}, see A002450 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 22 2005

For n>=2, a(n) is equal to the number of functions f:{1,2,...,n+1}->{1,2,3} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3} we have f(x_1)<>y_1 and f(x_2)<> y_2. - Milan R. Janjic (agnus(AT)blic.net), Apr 19 2007

A. M. Nemirovsky et al., Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108.

T. D. Noe, Table of n, a(n) for n=0..200

Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

D. J. Broadhurst, On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 305

Index entries for sequences related to trees

[4*3^(n-1)]. G.f.: (1+x)/(1-3x).

a(n) = Sum_{ 0<=k<=n } A029653(n, k)*x^k for x = 2 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jul 10 2005

The Hankel transform of this sequence is [1,-4,0,0,0,0,0,0,0,0,...] - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 21 2007

if n = 0 then 1 else 4*3^(n-1); fi;

with(combstruct):ZL0:=S=Prod(Sequence(Prod(a, Sequence(b))), b):ZL1:=Prod(begin_blockP, Z, end_blockP):ZL2:=Prod(begin_blockLR, Z, Sequence(Prod(mu_length, Z), card>=1), end_blockLR): ZL3:=Prod(begin_blockRL, Sequence(Prod(mu_length, Z), card>=1), Z, end_blockRL):Q:=subs([a=Union(ZL1, ZL2, ZL3), b=ZL1], ZL0), begin_blockP=Epsilon, end_blockP=Epsilon, begin_blockLR=Epsilon, end_blockLR=Epsilon, begin_blockRL=Epsilon, end_blockRL=Epsilon, mu_length=Epsilon:temp15:=draw([S, {Q}, unlabelled], size=15):seq(count([S, {Q}, unlabelled], size=n), n=2..27); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Mar 08 2008

(PARI) a(n)=if(n<1, n==0, 4*3^(n-1))

Sequence in context: A006817 A003119 A001394 this_sequence A052156 A000781 A055395

Adjacent sequences: A003943 A003944 A003945 this_sequence A003947 A003948 A003949

nonn,easy,nice,walk

njas

Additional comments from Michael Somos, Jun 18 2002