OFFSET
0,5
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Jean-Luc Baril and José L. Ramírez, Knight's paths towards Catalan numbers, Univ. Bourgogne Franche-Comté (2022).
Vaclav Kotesovec, Recurrence (of order 11)
J. Labelle and Y.-N. Yeh, Dyck paths of knight moves, Discrete Applied Math., 24 (1989), 213-221.
FORMULA
G.f.: z^2*A^2/(1-z*A), where A = (1+2*z+sqrt(1-4*z+4*z^2-4*z^4) -sqrt(2)*sqrt(1-4*z^2-2*z^4+(2*z+1)*sqrt(1-4*z+4*z^2-4*z^4)))/(4*z^2).
a(n) ~ c * (1+sqrt(3))^n / n^(3/2), where c = 4/sqrt(Pi*(27 + 17*sqrt(3) - sqrt(2*(730 + 929*sqrt(3))/3))) = 0.5480566813380593118... - Vaclav Kotesovec, Feb 29 2016
a(n) = Sum_{m=2..n} (m*Sum_{i=0..n-m }((Sum_{j=0..i+m }(binomial(i+m,j)*binomial(j,i-j)))*Sum_{k=0..n-i-m }((binomial(2*k+i+m-1,k)*Sum_{l=0..k}(binomial(k,l)*binomial(k-l,n-3*l-k-i-m)*(-1)^(n-l-k-m)))/(k+i+m)))). - Vladimir Kruchinin, Mar 06 2016
MATHEMATICA
a = (2*z + Sqrt[-4*z^4 + 4*z^2 - 4*z + 1] - Sqrt[2]*Sqrt[-2*z^4 - 4*z^2 + (2*z + 1)*Sqrt[-4*z^4 + 4*z^2 - 4*z + 1] + 1] + 1)/(4*z^2); gf = z^2*a^2/(1 - z*a); CoefficientList[Series[gf, {z, 0, 31}], z] (* Jean-François Alcover, Dec 21 2012, from g.f. *)
PROG
(Maxima)
a(n):=sum(m*sum((sum(binomial(i+m, j)*binomial(j, i-j), j, 0, i+m))*sum((binomial(2*k+i+m-1, k)*sum(binomial(k, l)*binomial(k-l, n-3*l-k-i-m)*(-1)^(n-l-k-m), l, 0, k))/(k+i+m), k, 0, n-i-m), i, 0, n-m), m, 2, n); /* Vladimir Kruchinin, Mar 06 2016 */
CROSSREFS
KEYWORD
nonn,easy,nice,walk
AUTHOR
EXTENSIONS
More terms from Emeric Deutsch, Dec 17 2003
STATUS
approved