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A054474 Number of walks on square lattice that start and end at origin after 2n steps, not touching origin at intermediate stages. 13
1, 4, 20, 176, 1876, 22064, 275568, 3584064, 47995476, 657037232, 9150655216, 129214858304, 1845409805168, 26606114089024, 386679996988736, 5658611409163008, 83302885723872852, 1232764004638179504, 18327520881735288432 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

1-dimensional and 3-dimensional analogs are A002420 and A049037.

Trajectories returning to the origin are prohibited, contrary to the situation in A094061.

REFERENCES

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 322-331.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..250

S. R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice [Broken link]

Steven R. Finch, Symmetric Random Walk on n-Dimensional Integer Lattice [Cached copy, with permission of the author]

FORMULA

G.f.: 2 - AGM(1, (1-16*x)^(1/2)).

G.f.: 2 - 1/hypergeom([1/2,1/2],[1],16*x). - Joerg Arndt, Jun 16 2011

Let (in Maple notation) G(x):=4/Pi*EllipticK(4*t)-2/Pi*EllipticF(1/sqrt(2+4*t), 4*t)-2/Pi*EllipticF(1/sqrt(2-4*t), 4*t), then GenFunc(x):=2-1/G(x). - Sergey Perepechko, Sep 11 2004

G.f.: 2-Pi/(2*EllipticK(4*sqrt(x))). - Vladeta Jovovic, Jun 23 2005

a(n) ~ Pi * 16^n / (n * log(n)^2) * (1 - (2*gamma + 8*log(2)) / log(n) + (3*gamma^2 + 24*log(2)*gamma + 48*log(2)^2 - Pi^2/2) / log(n)^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Sep 29 2019

EXAMPLE

a(5)=22064, i.e., there are 22064 different walks (on a square lattice) that start and end at origin after 2*5=10 steps, avoiding origin at intermediate steps.

MATHEMATICA

m = 18; gf[x_] = 2 - Pi/(2*EllipticK[4*Sqrt[x]]); (List @@ Normal[ Series[ gf[x], {x, 0, m-1}]] /. x -> 1)[[1 ;; m+1]]*Table[4^k, {k, 0, m}] (* Jean-Fran├žois Alcover, Jun 16 2011, after Vladeta Jovovic *)

CoefficientList[Series[2-Pi/(2*EllipticK[16*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Mar 10 2014 *)

PROG

(PARI) a(n)=if(n<0, 0, polcoeff(2-agm(1, sqrt(1-16*x+x*O(x^n))), n))

CROSSREFS

Cf. A002894, A002420, A049037.

Sequence in context: A068965 A185672 A210438 * A213144 A215873 A066917

Adjacent sequences:  A054471 A054472 A054473 * A054475 A054476 A054477

KEYWORD

easy,nonn,walk

AUTHOR

Alessandro Zinani (alzinani(AT)tin.it), May 19 2000

STATUS

approved

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Last modified January 23 19:12 EST 2020. Contains 331175 sequences. (Running on oeis4.)