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A197657 Row sums of A194595. 4
1, 4, 22, 134, 866, 5812, 40048, 281374, 2006698, 14482064, 105527060, 775113440, 5731756720, 42628923040, 318621793472, 2391808860446, 18023208400634, 136271601087352, 1033449449559724, 7858699302115444, 59906766929537116, 457685157123172664 (list; graph; refs; listen; history; text; internal format)
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.

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)^(2-j)/(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-((n-k)/(1+k))^3)/(1+2*k-n) if 1+2*k-n <> 0 else h(n,k) = 3. - Peter Luschny, Nov 24 2011

a(n) = A141147(n+1)/2 = A110707(n+1)/6 = (A000172(n)+A000172(n+1))/3. - Max Alekseyev, Jul 15 2014

Conjecture: (n+1)^2*a(n) -3*(n+1)*(2*n+1)*a(n-1) -3*n*(5*n-7)*a(n-2) -8*(n-2)^2*a(n-3)=0. - R. J. Mathar, Jul 26 2014

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

A197657 := proc(n)

    (A000172(n)+A000172(n+1))/3 ;

end proc; # R. J. Mathar, Jul 26 2014

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, 1-n, 1-n, 1-n], [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-((n-k)/(1+k))^3)/(1+2*k-n))*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

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Last modified November 22 20:13 EST 2014. Contains 249827 sequences.