OFFSET
1,1
COMMENTS
a(100)/6^93 = 33.333333333333370756088277230775... is the expected playing time of the "Snakes and Ladders" game on the empty board with all snakes and ladders removed. Althoen et al. (see link p. 74) cite this as "almost exactly 33 moves". One can assume that the omission of the addend of 1/3 was an obvious oversight.
LINKS
Hugo Pfoertner, Table of n, a(n) for n = 1..200
S. C. Althoen, L. King, K. Schilling, How long is a game of snakes and ladders? The Mathematical Gazette, Vol. 77, No. 478 (Mar., 1993), pp. 71-76.
FORMULA
Conjectures from Colin Barker, Feb 21 2020: (Start)
G.f.: x*(6 - 36*x - 36*x^2 - 36*x^3 - 36*x^4 - 36*x^5 - 35*x^6 + 279930*x^7 + 279900*x^8 + 279720*x^9 + 278640*x^10 + 272160*x^11 + 233280*x^12) / ((1 - 6*x)^2*(1 + 5*x + 24*x^2 + 108*x^3 + 432*x^4 + 1296*x^5)).
a(n) = 7*a(n-1) - 46656*a(n-7) for n>13.
(End)
PROG
(PARI) xpected(n, m)={my(M=matrix(n+1, n+1, i, j, 0)); for(i=1, n+1, my(kadd=0); for(j=i+1, i+m, if(j>n+1, kadd++, M[i, j]=1)); M[i, i]+=kadd); vecsum((1/(matid(n)-M[1..n, 1..n]/m))[1, ])};
for(k=1, 27, my(x=xpected(k, 6)); print1(numerator(x), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Feb 19 2020
STATUS
approved