

A331944


a(n)/ceiling(6^(n7)) is the expected number of rolls of a fair 6sided die in a game where the player starts at 0, advances the position by the outcome of the die's roll until exactly position n is reached. Positions beyond n are avoided by staying at the last visited position, but counting the rolls.


1



6, 6, 6, 6, 6, 6, 7, 43, 265, 1639, 10177, 63463, 397585, 2456503, 15189313, 93961351, 581260273, 3594003799, 22197096865, 136829952295, 843199062097, 5193720847351, 31972185139201, 196686016677319, 1209120275495089, 7428214177132183, 45613560985649761
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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. 7176.


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(n1)  46656*a(n7) 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

Cf. A000400, A127984.
Sequence in context: A113427 A082509 A226280 * A103337 A241155 A245399
Adjacent sequences: A331941 A331942 A331943 * A331945 A331946 A331947


KEYWORD

nonn


AUTHOR

Hugo Pfoertner, Feb 19 2020


STATUS

approved



