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A331944 a(n)/ceiling(6^(n-7)) is the expected number of rolls of a fair 6-sided 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 (list; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A113427 A082509 A226280 * A103337 A241155 A245399
KEYWORD
nonn
AUTHOR
Hugo Pfoertner, Feb 19 2020
STATUS
approved

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Last modified March 29 08:13 EDT 2024. Contains 371265 sequences. (Running on oeis4.)