A078679 -id:A078679 - cp's OEIS frontend

A192683 Floor-Sqrt transform of numbers of A078679 (Grand Motzkin paths with no zigzags).

1, 1, 1, 2, 4, 6, 10, 17, 27, 45, 74, 122, 201, 332, 548, 907, 1502, 2490, 4131, 6859, 11396, 18944, 31510, 52437, 87305, 145418, 242310, 403909, 673507, 1123409, 1874396, 3128270, 5222267, 8720050, 14563897, 24329283, 40650780, 67934677, 113551476, 189831651, 317406103, 530797959, 887783981

Offset: 0

Emanuele Munarini, Jul 07 2011

nonn

FSFromSeries[f_,x_,n_] := Map[Floor[Sqrt[#]]&,CoefficientList[Series[f,{x,0,n}],x]]
FSFromSeries[Sqrt[(1-x+x^2)/(1-3x+x^3+x^4)],x,100]

a(n) = floor(sqrt(A078679(n))).

A078678 Number of binary strings with n 1's and n 0's avoiding zigzags, that is avoiding the substrings 101 and 010.

Original entry on oeis.org

1, 2, 4, 8, 18, 42, 100, 242, 592, 1460, 3624, 9042, 22656, 56970, 143688, 363348, 920886, 2338566, 5949148, 15157874, 38674978, 98803052, 252701484, 646990518, 1658066668, 4252908542, 10917422860, 28046438252, 72099983802, 185469011130, 477383400300

Offset: 0

Emanuele Munarini, Dec 17 2002

nonn

Also number of Grand Dyck paths of length 2*n with no zigzags, that is, with no factors UDU or DUD. - Emanuele Munarini, Jul 07 2011

			For n = 2 : 0011, 0110, 1001, 1100.
For n = 3 : 000111, 011001, 100011, 110001, 001110, 011100, 100110, 111000.

Vincenzo Librandi, Table of n, a(n) for n = 0..220
Andrei Asinowski and Cyril Banderier, On Lattice Paths with Marked Patterns: Generating Functions and Multivariate Gaussian Distribution, 31st International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2020) Leibniz International Proceedings in Informatics (LIPIcs) Vol. 159, 1:1-1:16.
T. Doslic, Seven lattice paths to log-convexity, Acta Appl. Mathem. 110 (3) (2010) 1373-139, eq 4.
Emanuele Munarini and N. Z. Salvi, Binary strings without zigzags, Séminaire Lotharingien de Combinatoire, B49h (2004), 15 pp.
R. Pemantle and M. C. Wilson, Twenty Combinatorial Examples of Asymptotics Derived from Multivariate Generating Functions, arXiv:math/0512548 [math.CO], 2007.

Cf. A078679, A128588.
Cf. A003440.
Main diagonal of array A099172.
Related to diagonal of rational functions: A268545-A268555.

a:= proc(n) option remember; `if`(n<5, [1, 2, 4, 8, 18][n+1],
     (2*n*a(n-1)+(n-2)*a(n-2)+(2*n-8)*a(n-3)-(n-4)*a(n-4))/n)
    end:
seq(a(n), n=0..40);  # Alois P. Heinz, Feb 13 2020

Table[SeriesCoefficient[Series[Sqrt[(1 + x + x^2)/(1 - 3 x + x^2)], {x, 0, n}], n], {n, 0, 40}]

a(n):=coeff(taylor((1+x+x^2)/sqrt(1-2*x-x^2-2*x^3+x^4),x,0,n),x,n);
makelist(a(n),n,0,12); /* Emanuele Munarini, Jul 07 2011 */

my(x='x+O('x^99)); Vec(((1+x+x^2)/(1-3*x+x^2))^(1/2)) \\ Altug Alkan, Jul 18 2016

G.f.: sqrt( ( 1 + x + x^2 ) / ( 1 - 3*x + x^2 ) ).
a(n) = Sum_{k=0..n+floor(n/2)} binomial( n - k + 2*floor(k/3), floor(k/3) )^2.
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k,k)^2*( 2*n^2 - 6*n*k + 6*k^2 )/(n-k)^2, n > 0.
a(n) ~ 2 * ((3+sqrt(5))/2)^n / (5^(1/4)*sqrt(Pi*n)). - Vaclav Kotesovec, Mar 21 2014
a(n) = [x^n y^n](1+x*y+x^2*y^2)/(1-x-y+x*y-x^2*y^2). - Gheorghe Coserea, Jul 18 2016
D-finite with recurrence: n*a(n) -2*n*a(n-1) +(-n+2)*a(n-2) +2*(-n+4)*a(n-3) +(n-4)*a(n-4)=0. [Doslic] - R. J. Mathar, Jun 21 2018

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A192683 Floor-Sqrt transform of numbers of A078679 (Grand Motzkin paths with no zigzags).

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A078678 Number of binary strings with n 1's and n 0's avoiding zigzags, that is avoiding the substrings 101 and 010.

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