



1, 4, 22, 134, 866, 5812, 40048, 281374, 2006698, 14482064, 105527060, 775113440, 5731756720, 42628923040, 318621793472, 2391808860446, 18023208400634, 136271601087352, 1033449449559724, 7858699302115444, 59906766929537116, 457685157123172664
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OFFSET

0,2


COMMENTS

Number of meanders of length (n+1)*3 which are composed by arcs of equal length and a central angle of 120 degrees.
Definition of a meander:
A binary curve C is a triple (m, S, dir) such that
(a) S is a list with values in {L,R} which starts with an L,
(b) dir is a list of m different values, each value of S being allocated a value of dir,
(c) consecutive Ls increment the index of dir,
(d) consecutive Rs decrement the index of dir,
(e) the integer m>0 divides the length of S and
(f) C is a meander if each value of dir occurs length(S)/m times.
For this sequence, m = 3.
The terms are proved by brute force for 0 <= n <= 11, but not yet in general. [Susanne Wienand, Oct 29 2011]


LINKS

Table of n, a(n) for n=0..21.
Peter Luschny, Meanders and walks on the circle.
Susanne Wienand, Animation of a meander
Susanne Wienand, Example of a meander


FORMULA

a(n) = Sum{k=0..n} Sum{j=0..2} Sum{i=0..2} (1)^(j+i)*C(i,j)*C(n,k)^3*(n+1)^j*(k+1)^(2j)/(k+1)^2.  Peter Luschny, Nov 02 2011
a(n) = Sum_{k=0..n} h(n,k)*binomial(n,k)^3, where h(n,k) = (1+k)*(1((nk)/(1+k))^3)/(1+2*kn) if 1+2*kn <> 0 else h(n,k) = 3.  Peter Luschny, Nov 24 2011


EXAMPLE

Some examples of list S and allocated values of dir if n = 4:
Length(S) = (4+1)*3 = 15.
S: L,L,L,L,L,L,L,L,L,L,L,L,L,L,L
dir: 1,2,0,1,2,0,1,2,0,1,2,0,1,2,0
S: L,L,L,L,R,L,L,R,L,L,R,L,L,L,L
dir: 1,2,0,1,1,1,2,2,2,0,0,0,1,2,0
S: L,R,L,L,L,L,L,R,L,L,R,L,R,R,R
dir: 1,1,1,2,0,1,2,2,2,0,0,0,0,2,1
Each value of dir occurs 15/3 = 5 times.


MAPLE

# Floating point evaluation! Increase DIG for larger n.
alias(HG = hypergeom); DIG := 32; R := n > HG([n, n, n], [2, 2], 1)*(1 + n + n^2) + HG([1  n, 1  n, 1  n], [3, 3], 1)*(n^3  n^4)/4 + HG([2, 1  n, 1  n, 1  n], [1, 3, 3], 1)*n^3/4; seq(round(evalf(R(n), DIG)), n=0..21); # Peter Luschny, Oct 21 2011


MATHEMATICA

A197657[n_] := Sum[Sum[Sum[(1)^(j + i)* Binomial[i, j]*Binomial[n, k]^3*(n + 1)^j*(k + 1)^(2  j)/(k + 1)^2, {i, 0, 2}], {j, 0, 2}], {k, 0, n}]; Table[A197657[n], {n, 0, 16}] (* Peter Luschny, Nov 02 2011 *)


PROG

(SAGE) from mpmath import *
def A197657(n) : return hyper([n, n, n], [2, 2], 1)*(1 + n + n^2) + hyper([1  n, 1  n, 1  n], [3, 3], 1)*(n^3  n^4)/4 + hyper([2, 1n, 1n, 1n], [1, 3, 3], 1)*n^3/4
mp.dps = 32
for n in (0..21) : print A197657(n) ## Peter Luschny, Oct 24 2011
(PARI)
A197657(n) = {sum(k=0, n, if(n == 1+2*k, 3, (1+k)*(1((nk)/(1+k))^3)/(1+2*kn))*binomial(n, k)^3)} \\ Peter Luschny, Nov 24 2011


CROSSREFS

Cf. A198060, A198256, A198257, A198258.
Sequence in context: A180899 A007195 A193620 * A183280 A183281 A067120
Adjacent sequences: A197654 A197655 A197656 * A197658 A197659 A197660


KEYWORD

nonn


AUTHOR

Susanne Wienand, Oct 17 2011


STATUS

approved



