login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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: A007195 A193620 A274745 * A183280 A183281 A067120

Adjacent sequences:  A197654 A197655 A197656 * A197658 A197659 A197660

KEYWORD

nonn

AUTHOR

Susanne Wienand, Oct 17 2011

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 10 21:23 EST 2016. Contains 279011 sequences.