A198258 Row sums of A197655.

1, 7, 190, 5831, 219626, 8976562, 394800204, 18211045575, 873216720082, 43136486269382, 2183722698469676, 112795257850321202, 5925951358743662260, 315869014732813229716, 17048301919464100932440, 930217336628892162216135, 51244644190683748419791970

Offset

0,2

Comments

Number of meanders of length (n+1)*6 which are composed by arcs of equal length and a central angle of 60 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 = 6.

The terms are proved by brute force for 0 <= n <= 5, but not yet in general. - Susanne Wienand, Oct 29 2011

Links

Table of n, a(n) for n=0..16.

Peter Luschny, Meanders and walks on the circle.

Formula

a(n) = Sum_{k=0..n} Sum_{j=0..5} Sum_{i=0..5} (-1)^(j+i)*C(i,j)*C(n,k)^6*(n+1)^j*(k+1)^(5-j)/(k+1)^5. - Peter Luschny, Nov 02 2011

a(n) = Sum_{k=0..n} h(n,k)*binomial(n,k)^6, where h(n,k) = (1+k)*(1-((n-k)/(1+k))^6)/(1+2*k-n) if 1+2*k-n <> 0 else h(n,k) = 6. - Peter Luschny, Nov 24 2011

Conjecture: working with offset 1, that is, a(1) = 1, a(2) = 7, ..., then the supercongruence a(n*p^r) == a(n*p^(r-1)) (mod p^(3*r)) holds for positive integers n and r and all primes p >= 3. - Peter Bala, Mar 21 2023

a(n) ~ sqrt(3) * 2^(6*n+3) / (Pi*n)^(5/2). - Vaclav Kotesovec, Apr 17 2023

Example

Some examples of list S and allocated values of dir if n = 4:
Length(S) = (4+1)*6 = 30.
  S: L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L,L
dir: 1,2,3,4,5,0,1,2,3,4,5,0,1,2,3,4,5,0,1,2,3,4,5,0,1,2,3,4,5,0
  S: L,L,L,L,L,L,L,R,R,R,R,L,R,R,R,R,R,L,L,L,L,R,R,L,L,L,L,R,L,L
dir: 1,2,3,4,5,0,1,1,0,5,4,4,4,3,2,1,0,0,1,2,3,3,2,2,3,4,5,5,5,0
  S: L,L,L,L,L,R,R,R,R,R,R,R,L,L,R,L,L,R,L,L,R,L,L,L,R,R,L,L,L,L
dir: 1,2,3,4,5,5,4,3,2,1,0,5,5,0,0,0,1,1,1,2,2,2,3,4,4,3,3,4,5,0
Each value of dir occurs 30/6 = 5 times.

Maple

A198258 := proc(n) local i, j, k, pow;
pow := (a, b) -> if a=0 and b=0 then 1 else a^b fi;
add(add(add((-1)^(j+i)*binomial(i, j)*binomial(n, k)^6*pow(n+1, j)*pow(k+1, 5-j)/(k+1)^5, i=0..5), j=0..5), k=0..n) end:
seq(A198258(n), n=0..16); # Peter Luschny, Nov 02 2011

Mathematica

T[n_, k_] := (1 + n)(1 + 3 k + 3 k^2 - n - 3 k*n + n^2)(1 + k + k^2 + n - k*n + n^2) Binomial[n, k]^6/(1 + k)^5;
a[n_] := Sum[T[n, k], {k, 0, n}];
Table[a[n], {n, 0, 16}] (* Jean-François Alcover, Jun 29 2019 *)

Prog

(PARI)
A198258(n) = {sum(k=0, n, if(n == 1+2*k, 6, (1+k)*(1-((n-k)/(1+k))^6)/(1+2*k-n))*binomial(n, k)^6)} \\ Peter Luschny, Nov 24 2011

Crossrefs

Cf. A198060, A198256, A198257.

Sequence in context: A303292 A010332 A368592 * A200819 A232146 A264353

Adjacent sequences: A198255 A198256 A198257 * A198259 A198260 A198261

Keyword

nonn

Author

Susanne Wienand, Oct 24 2011

Status

approved