login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A005559 Number of walks on square lattice.
(Formerly M1832)
3
1, 2, 8, 20, 75, 210, 784, 2352, 8820, 27720, 104544, 339768, 1288287, 4294290, 16359200, 55621280, 212751396, 734959368, 2821056160, 9873696560, 38013731756, 134510127752, 519227905728, 1854385377600, 7174705330000, 25828939188000, 100136810390400 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

REFERENCES

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Table of n, a(n) for n=1..27.

R. K. Guy, Letter to N. J. A. Sloane, May 1990

R. K. Guy, Catwalks, sandsteps and Pascal pyramids, J. Integer Sequences, Vol. 3 (2000), Article #00.1.6 (see figure 5).

FORMULA

a(n) = C(n+1,ceiling((n-1)/2)) *C(n,floor((n-1)/2)) -C(n+1,ceiling((n-2)/2)) *C(n,floor((n-2)/2)). - Paul D. Hanna, Apr 16 2004

G.f.: -(48*x^3-16*x^2-3*x+1)*EllipticK(4*x)/(12*Pi*x^4)+(4*x^2-9*x+1)*EllipticE(4*x)/(12*Pi*x^4)+1/(4*x^3)-1/(2*x^2) (using Maple's convention for elliptic integrals: EllipticE(t) = int_0^1 sqrt(1 - s^2*t^2)/sqrt(1-s^2) ds, EllipticK(t) = int_0^1 ((1-s^2*t^2)*(1-s^2))^(-1/2) ds). - Robert Israel, Oct 19 2014

Conjecture: -(n-1)*(2*n+1)*(n+4)*(n+3)*a(n) +4*(n+1)*(2*n^2+4*n+9)*a(n-1) +16*n*(n-1)*(2*n+3)*(n+1)*a(n-2)=0. - R. J. Mathar, Apr 02 2017

MAPLE

seq(binomial(n+1, ceil((n-1)/2))*binomial(n, floor((n-1)/2)) -binomial(n+1, ceil((n-2)/2))*binomial(n, floor((n-2)/2)), n=1..30); # Robert Israel, Oct 19 2014

MATHEMATICA

Table[Binomial[n+2, Ceiling[n/2]] Binomial[n+1, Floor[n/2]] - Binomial[n+2, Ceiling[(n-1)/2]] Binomial[n+1, Floor[(n-1)/2]], {n, 0, 200}] (* Vincenzo Librandi, Oct 17 2014 *)

PROG

(PARI) {a(n)=binomial(n+2, ceil(n/2))*binomial(n+1, floor(n/2)) - binomial(n+2, ceil((n-1)/2))*binomial(n+1, floor((n-1)/2))}

(MAGMA) [Binomial(n+2, Ceiling(n/2))*Binomial(n+1, Floor(n/2)) - Binomial(n+2, Ceiling((n-1)/2))*Binomial(n+1, Floor((n-1)/2)): n in [0..30]]; // Vincenzo Librandi, Oct 16 2014

CROSSREFS

Cf. A005558-A005560, A093768.

Sequence in context: A100097 A133467 A091004 * A001471 A162585 A000159

Adjacent sequences:  A005556 A005557 A005558 * A005560 A005561 A005562

KEYWORD

nonn,walk

AUTHOR

N. J. A. Sloane

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 March 21 22:19 EDT 2019. Contains 321382 sequences. (Running on oeis4.)