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A254129 Number of 2n-move closed knight paths on an unbounded chessboard from a given square to the same square. 7
1, 8, 168, 5840, 261800, 13180608, 702273264, 38641656768, 2171652448680, 123938999632448, 7158206751686848, 417418594698260064, 24535017440445455216, 1451786144317963971200, 86396682439552099487040, 5166936574734171792925440, 310340697572034456203934120 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Every move changes the color of the square the knight is on, so there is no returning path of odd length.

LINKS

David A. Corneth and Alois P. Heinz, Table of n, a(n) for n = 0..550 (first 85 terms from David A. Corneth)

Shalosh B. Ekhad and Doron Zeilberger, In How Many Ways Can the Chess Pieces Walk n Steps, Staying on the Board?, May 19 2011

Mohamud Mohammed and Doron Zeilberger, Maple program SMAZ

Doron Zeilberger, Input file inSMAZ6

Doron Zeilberger, Output from Maple program SMAZ

FORMULA

From Robert Israel, Jan 26 2015: (Start)

a(n) = 4^n * (2*Pi)^(-2) * int_0^(2*Pi) int_0^(2*Pi) ds dt (cos(s+2*t)+cos(s-2*t)+cos(2*s+t)+cos(2*s-t))^(2*n).

G.f.: (2*Pi)^(-2) * int_0^(2*Pi) int_0^(2*Pi) ds dt 1/(1 - 4*x*(cos(s+2*t)+cos(s-2*t)+cos(2*s+t)+cos(2*s-t))^2).

(End)

a(n) ~ 64^n / (5*Pi*n). - Vaclav Kotesovec, Jan 28 2015

Recurrence (conjectured): 3*n^2*(3*n-2)*(3*n-1)*(4*n-3)*(4*n-1)*(58625*n^6 - 574525*n^5 + 2317575*n^4 - 4929815*n^3 + 5836090*n^2 - 3647730*n + 940788)*a(n) = 4*(563444875*n^12 - 7212094400*n^11 + 40894216825*n^10 - 135653664390*n^9 + 292742658975*n^8 - 432166599360*n^7 + 446527351283*n^6 - 324481592710*n^5 + 164046706898*n^4 - 56035458036*n^3 + 12203976528*n^2 - 1507156200*n + 78246000)*a(n-1) - 64*(n-1)*(2*n-3)^2*(167726125*n^9 - 1643716025*n^8 + 6735239425*n^7 - 15048594215*n^6 + 20072439970*n^5 - 16473493280*n^4 + 8273936628*n^3 - 2437948332*n^2 + 377982648*n - 22556880)*a(n-2) + 165888*(n-2)*(n-1)*(2*n - 5)^2*(2*n - 3)^2*(58625*n^6 - 222775*n^5 + 324325*n^4 - 232265*n^3 + 86220*n^2 - 15570*n + 1008)*a(n-3). - Vaclav Kotesovec, Jan 30 2015

a(n) = the constant term in the expansion of (x^2*y + x*y^2 + y^2/x + y/x^2 + 1/(x^2*y) + 1/(y^2*x) + x/y^2 + x^2/y)^(2*n). - Peter Bala, Feb 14 2017

The conjectured recurrence of Vaclav Kotesovec is true. Running the input file inSMAZ6 (see Links) on the Maple program SMAZ gives the recurrence followed by the certificate shown in the output file oSMAZ6.  - Doron Zeilberger, Mar 29 2019

EXAMPLE

a(1) = 8. For illustration, let's assume we're on a usual 8 X 8 chessboard, with the knight initially at D4. Then there are 8 paths bringing it back to D4 in 2 moves:

D4-E6-D4; D4-F5-D4; D4-F3-D4; D4-E2-D4; D4-C2-D4; D4-B3-D4; D4-B5-D4; D4-C6-D4.

MAPLE

G:= cos(x+2*y)+cos(x-2*y)+cos(2*x+y)+cos(2*x-y):

F:= 1: a[0]:= 1:

for n from 1 to 20 do

  F:= combine(F*G^2, trig);

  a[n]:= 4^n*remove(has, F, cos);

od:

seq(a[n], n=0..20);  # Robert Israel, Jan 26 2015

# second Maple program:

b:= proc(n, x, y) option remember; `if`(max(x, y)>2*n or x+y>3*n, 0,

      `if`(n=0, 1, add(b(n-1, abs(x+l[1]), abs(y+l[2])), l=[[1, 2],

      [2, 1], [-1, 2], [-2, 1], [1, -2], [2, -1], [-1, -2], [-2, -1]])))

    end:

a:= n-> b(2*n, 0$2):

seq(a(n), n=0..25);  # Alois P. Heinz, Jan 29 2015

MATHEMATICA

b[n_, x_, y_] := b[n, x, y] = If[Max[x, y] > 2n || x+y > 3n, 0, If[n == 0, 1, Sum[b[n-1, Abs[x+l[[1]]], Abs[y+l[[2]]]], {l, {{1, 2}, {2, 1}, {-1, 2}, {-2, 1}, {1, -2}, {2, -1}, {-1, -2}, {-2, -1}}}]]];

a[n_] :=  b[2n, 0, 0];

Table[a[n], {n, 0, 25}] (* Jean-Fran├žois Alcover, Apr 30 2019, after Alois P. Heinz *)

PROG

(PARI) a(n)={my(l=listcreate(), v=vector(2*n+1)); m=matrix(1, 1); m[1, 1]=1; listput(l, m); v[1]=1; for(i=2, 2*n+1, m=matrix(4*i-3, 4*i-3); for(j=1, #l[i-1], for(k=1, #l[i-1], m[j+2-2, k+2-1]+=l[i-1][j, k]; m[j+2-2, k+2+1]+=l[i-1][j, k]; m[j+2-1, k+2-2]+=l[i-1][j, k]; m[j+2-1, k+2+2]+=l[i-1][j, k]; m[j+2+1, k+2-2]+=l[i-1][j, k]; m[j+2+1, k+2+2]+=l[i-1][j, k]; m[j+2+2, k+2-1]+=l[i-1][j, k]; m[j+2+2, k+2+1]+=l[i-1][j, k])); v[i]=m[2*i-1, 2*i-1]; listput(l, m); ); listput(l, v); v[#v]} \\ David A. Corneth, Jan 26 2015

(PARI) {a(n) = polcoef(polcoef((x^2*y+x*y^2+y^2/x+y/x^2+1/(x^2*y)+1/(x*y^2)+x/y^2+x^2/y)^(2*n), 0), 0)} \\ Seiichi Manyama, Nov 02 2019

CROSSREFS

Cf. A025600, A094061, A253974, A254459, A307347.

Sequence in context: A307347 A253974 A254459 * A334780 A084941 A139564

Adjacent sequences:  A254126 A254127 A254128 * A254130 A254131 A254132

KEYWORD

nonn,easy

AUTHOR

David A. Corneth, Jan 25 2015

STATUS

approved

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Last modified July 2 11:54 EDT 2020. Contains 335398 sequences. (Running on oeis4.)