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.

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

Offset: 1

Author

Hugo Pfoertner, Feb 19 2020

Keywords

• nonn

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.

Crossrefs

Cf. A000400, A127984.

Programs

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),", "))

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)