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

 

Logo

The OEIS is looking to hire part-time people to help edit core sequences, upload scanned documents, process citations, fix broken links, etc. - Neil Sloane, njasloane@gmail.com

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A025597 Number of n-move king paths on 8 X 8 board from given corner to opposite corner. 1
0, 0, 0, 0, 0, 0, 0, 1, 56, 1309, 20370, 255366, 2782296, 27630317, 256617790, 2269878170, 19345170656, 160223380546, 1297456951652, 10319966008680, 80906898257760, 626886465395595, 4810654849509082, 36623649326935517 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,9

COMMENTS

As long as the path starts and ends on the correct square, it is counted. The path may revisit squares, including the start and end squares.

LINKS

David W. Wilson and Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 49 terms from David W. Wilson)

FORMULA

a(n) = (4/81) * Sum_{j,k=1..8} (-1)^(j+k)* [sin(j*Pi/9)*sin(k*Pi/9)]^2 *[(1+2*cos(j*Pi/9))*(1+2*cos(k*Pi/9))-1]^n. - Andrew G. Buchanan, Jun 24 2012

G.f.: -(408459*x^21 +3108249*x^20 +5135985*x^19 -8733022*x^18 -29723403*x^17 -6771900*x^16 +52706117*x^15 +58351590*x^14 -6069219*x^13 -51965240*x^12 -37661505*x^11 -6328524*x^10 +5718540*x^9 +3500727*x^8 +471552*x^7 -208258*x^6 -90243*x^5 -9609*x^4 +1531*x^3 +498*x^2 +42*x+1) *x^7 / ((3*x-1) *(x+1) *(3*x^3-3*x-1) *(x^3-3*x+1) *(17*x^3+6*x^2-3*x-1) *(x^3+3*x^2-6*x+1) *(3*x^3+9*x^2+6*x-1) *(19*x^3-9*x^2-3*x+1) *(x^3+9*x^2+6*x+1) *(3*x^3-9*x^2-3*x+1) *(17*x^3+18*x^2+3*x-1)). - Alois P. Heinz, Jun 25 2012

EXAMPLE

The king cannot reach the opposite corner in fewer than 7 moves, hence a(0) through a(6) are all 0.

There is only one way to reach the opposite corner in 7 moves, namely along the main diagonal, so a(7) = 1.

MAPLE

b:= proc(n, i, j) option remember;

      `if`(n<0 or i<0 or i>7 or j<0 or j>7, 0, `if`([n, i, j]=[0$3],

       1, add(b(n-1, i+r[1], j+r[2]), r=[[1, 1], [1, 0], [1, -1],

          [0, 1], [0, -1], [-1, 1], [-1, 0], [-1, -1]])))

    end:

a:= n-> b(n, 7, 7):

seq(a(n), n=0..30);  # Alois P. Heinz, Jun 25 2012

MATHEMATICA

f[n_] := Round[4/81*Sum[(-1)^(j + k)*Sin[j*Pi/9]^2 Sin[k*Pi/9]^2*((1 + 2Cos[j*Pi/9])*(1 + 2Cos[k*Pi/9]) - 1)^n, {j, 8}, {k, 8}]]; Array[f, 23] (* Robert G. Wilson v, Jun 28 2012 *)

b[n_, i_, j_] := b[n, i, j] = If[n<0 || i<0 || i>7 || j<0 || j>7, 0, If[{n, i, j} == {0, 0, 0}, 1, Sum [b[n-1, i+r[[1]], j+r[[2]]], {r, {{1, 1}, {1, 0}, {1, -1}, {0, 1}, {0, -1}, {-1, 1}, {-1, 0}, {-1, -1}}}]]]; a[n_] := b[n, 7, 7]; Table[a[n], {n, 0, 30}] (* Jean-Fran├žois Alcover, May 28 2015, after Alois P. Heinz *)

CROSSREFS

Sequence in context: A183587 A219469 A278079 * A034202 A160290 A030649

Adjacent sequences:  A025594 A025595 A025596 * A025598 A025599 A025600

KEYWORD

nonn,easy

AUTHOR

David W. Wilson

EXTENSIONS

a(24)-a(48) from Wouter Meeussen, Jun 24 2012

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 | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified May 30 00:42 EDT 2017. Contains 287304 sequences.