PS
17 November 2021
I was able to find recurrences for the minimum number of pips
p(n) = 3*p(n-1) + 3*p(n-2) + 4*p(n-3) + 246, with p(1) = 1,
p(2) = 19, p(3) = 173 and p(4) = 734,
and for the maximum number of pips
m(n) = 3*m(n-1) + 3*m(n-2) + 4*m(n-3) - 246, with m(1) = 23,
m(2) = 77, m(3) = 211 and m(4) = 802.
Fausto Morales showed much better thinking (as always) and found closed formulas for the minimum number of pips
p(n) = 3*(2^2n) - 5*(2^3 - 2^((n-1) mod 3)) + 1, with p(1) = 1,
and for the maximum number of pips
m(n) = 3*(2^2n) + 5*(2^3 - 2^((n-1) mod 3)) - 1, with m(1) = 23.