cp's OEIS Frontend

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.

%I A331944 #19 Feb 12 2021 12:04:40
%S A331944 6,6,6,6,6,6,7,43,265,1639,10177,63463,397585,2456503,15189313,
%T A331944 93961351,581260273,3594003799,22197096865,136829952295,843199062097,
%U A331944 5193720847351,31972185139201,196686016677319,1209120275495089,7428214177132183,45613560985649761
%N 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.
%C A331944 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.
%H A331944 Hugo Pfoertner, <a href="/A331944/b331944.txt">Table of n, a(n) for n = 1..200</a>
%H A331944 S. C. Althoen, L. King, K. Schilling, <a href="https://doi.org/10.2307/3619261">How long is a game of snakes and ladders?</a> The Mathematical Gazette, Vol. 77, No. 478 (Mar., 1993), pp. 71-76.
%F A331944 Conjectures from _Colin Barker_, Feb 21 2020: (Start)
%F A331944 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)).
%F A331944 a(n) = 7*a(n-1) - 46656*a(n-7) for n>13.
%F A331944 (End)
%o A331944 (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,])};
%o A331944 for(k=1,27,my(x=xpected(k,6));print1(numerator(x),", "))
%Y A331944 Cf. A000400, A127984.
%K A331944 nonn
%O A331944 1,1
%A A331944 _Hugo Pfoertner_, Feb 19 2020