login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 56th year. In the past year we added 10000 new sequences and reached almost 9000 citations (which often say "discovered thanks to the OEIS").
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A198256 Row sums of A197653. 6
1, 5, 46, 485, 5626, 69062, 882540, 11614437, 156343330, 2142556130, 29791689148, 419260001030, 5960334608788, 85469709312860, 1234797737654296, 17955907741675749, 262607675818816050, 3860239468267647914, 57002176852356800700, 845159480056345448610 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Number of meanders of length (n+1)*4 which are composed by arcs of equal length and a central angle of 90 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 = 4.

The terms are proved by brute force for 0 <= n <= 8, but not yet in general. [Susanne Wienand, Oct 29 2011]

LINKS

Table of n, a(n) for n=0..19.

Peter Luschny, Meanders and walks on the circle.

FORMULA

a(n) = Sum{k=0..n} Sum{j=0..3} Sum{i=0..3} (-1)^(j+i)*C(i,j)*C(n,k)^4*(n+1)^j*(k+1)^(3-j)/(k+1)^3. - Peter Luschny, Nov 02 2011

a(n) = Sum_{k=0..n} h(n,k)*binomial(n,k)^4, where h(n,k) = (1+k)*(1-((n-k)/(1+k))^4)/(1+2*k-n) if 1+2*k-n <> 0 else h(n,k) = 4. - Peter Luschny, Nov 24 2011

EXAMPLE

Some examples of list S and allocated values of dir if n = 4:

Length(S) = (4+1)*4 = 20.

  S: L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L

dir: 1,2,3,0,1,2,3,0,1,2,3,0,1,2,3,0,1,2,3,0

  S: L,L,L,L,R,L,R,R,L,R,R,R,L,R,L,L,R,L,L,L

dir: 1,2,3,0,0,0,0,3,3,3,2,1,1,1,1,2,2,2,3,0

  S: L,L,R,L,L,L,L,R,R,L,R,R,L,R,L,L,L,L,R,R

dir: 1,2,2,2,3,0,1,1,0,0,0,3,3,3,3,0,1,2,2,1

Each value of dir occurs 20/4 = 5 times.

MATHEMATICA

A198256[n_] := Sum[Sum[Sum[(-1)^(j + i)* Binomial[i, j]*Binomial[n, k]^4*(n + 1)^j*(k + 1)^(3 - j)/(k + 1)^3, {i, 0, 3}], {j, 0, 3}], {k, 0, n}]; Table[A198256[n], {n, 0, 16}] (* Peter Luschny, Nov 02 2011 *)

PROG

(Sage)

from mpmath import mp, hyper

def A198256(n) : return hyper([1-n, 1-n, 1-n, 1-n], [3, 3, 3], 1)*(n^4-n^6)/4 + hyper([-n, -n, -n, -n], [2, 2, 2], 1)*(1+n+n^2+n^3) + hyper([2, 1-n, 1-n, 1-n, 1-n], [1, 3, 3, 3], 1)*(n^4+n^5)/4

mp.dps = 32

for n in (0..19) : print(int(A198256(n)))  # Peter Luschny, Oct 24 2011

(PARI)

A198256(n) = {sum(k=0, n, if(n == 1+2*k, 4, (1+k)*(1-((n-k)/(1+k))^4)/(1+2*k-n))*binomial(n, k)^4)} \\ Peter Luschny, Nov 24 2011

CROSSREFS

Cf. A198060, A198257, A198258.

Sequence in context: A042533 A309185 A169963 * A232972 A127304 A112029

Adjacent sequences:  A198253 A198254 A198255 * A198257 A198258 A198259

KEYWORD

nonn

AUTHOR

Susanne Wienand, Oct 22 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
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified November 29 19:26 EST 2020. Contains 338769 sequences. (Running on oeis4.)