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%I #51 Jul 12 2023 11:05:56
%S 1,6,94,1700,35466,795312,18848992,464517468,11801240050,307073982116,
%T 8147186436324,219664321959524,6003343077661216,165975724832822400,
%U 4634768975107569024,130553813782898706908,3705740233107582161538,105902829964290241990332
%N Row sums of A197654.
%C Number of meanders of length (n+1)*5 which are composed by arcs of equal length and a central angle of 72 degrees.
%C Definition of a meander:
%C A binary curve C is a triple (m, S, dir) such that
%C (a) S is a list with values in {L,R} which starts with an L,
%C (b) dir is a list of m different values, each value of S being allocated a value of dir,
%C (c) consecutive Ls increment the index of dir,
%C (d) consecutive Rs decrement the index of dir,
%C (e) the integer m>0 divides the length of S and
%C (f) C is a meander if each value of dir occurs length(S)/m times.
%C For this sequence, m = 5.
%C The terms are proved by brute force for 0 <= n <= 6, but not yet in general. [_Susanne Wienand_, Oct 29 2011]
%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/Meander">Meanders and walks on the circle</a>.
%H Project Euler, <a href="https://projecteuler.net/problem=208">Robot Walks: Problem 208</a>
%F a(n) = Sum{k=0..n} Sum{j=0..4} Sum{i=0..4} (-1)^(j+i)*C(i,j)*C(n,k)^5*(n+1)^j*(k+1)^(4-j)/(k+1)^4. - _Peter Luschny_, Nov 02 2011
%F a(n) = Sum_{k=0..n} h(n,k)*binomial(n,k)^5, where h(n,k) = (1+k)*(1-((n-k)/(1+k))^5)/(1+2*k-n) if 1+2*k-n <> 0 else h(n,k) = 5. - _Peter Luschny_, Nov 24 2011
%F a(n) ~ sqrt(5) * 2^(5*n+2) / (Pi*n)^2. - _Vaclav Kotesovec_, Apr 17 2023
%e Some examples of list S and allocated values of dir if n = 5:
%e Length(S) = (5+1)*5 = 30.
%e 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
%e dir: 1,2,3,4,0,1,2,3,4,0,1,2,3,4,0,1,2,3,4,0,1,2,3,4,0,1,2,3,4,0
%e S: L,L,L,L,L,L,L,L,L,L,L,L,L,R,L,R,R,R,R,R,L,R,L,L,L,L,R,R,R,L
%e dir: 1,2,3,4,0,1,2,3,4,0,1,2,3,3,3,3,2,1,0,4,4,4,4,0,1,2,2,1,0,0
%e S: L,L,L,L,L,R,L,L,L,R,R,L,L,L,L,L,R,R,L,R,R,L,R,R,L,L,L,L,L,R
%e dir: 1,2,3,4,0,0,0,1,2,2,1,1,2,3,4,0,0,4,4,4,3,3,3,2,2,3,4,0,1,1
%e Each value of dir occurs 30/5 = 6 times.
%p A198257 := proc(n) local i, j, k, pow;
%p pow := (a, b) -> if a=0 and b=0 then 1 else a^b fi;
%p add(add(add((-1)^(j+i)*binomial(i,j)*binomial(n,k)^5*pow(n+1,j)*pow(k+1,4-j)/(k+1)^4, i=0..4),j=0..4),k=0..n) end: seq(A198257(n), n=0..16); # _Peter Luschny_, Nov 02 2011
%t Table[Sum[Sum[ Sum[(-1)^(j + i) Binomial[i, j], {i, 0, 4}] Binomial[n, k]^5*(n + 1)^j*(k + 1)^(4 - j), {j, 0, 4}]/(k + 1)^4, {k, 0, n}], {n, 0, 17}] (* _Michael De Vlieger_, Aug 18 2016 *)
%o (PARI)
%o A198257(n) = {sum(k=0,n,if(n == 1+2*k,5,(1+k)*(1-((n-k)/(1+k))^5)/(1+2*k-n))*binomial(n,k)^5)} \\ _Peter Luschny_, Nov 24 2011
%Y Cf. A198060, A198256, A198258.
%K nonn
%O 0,2
%A _Susanne Wienand_, Oct 22 2011
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