I just finished exploring the 3x3x3 out to 12 face turns. Here are the statistics, first for positions with exactly that distance:
Next for positions with a distance less than or equal to d:
I believe this is the first time these numbers have been computed.
Code:
d mod M + inv mod M positions
-- ------------ ------------ --------------
0 1 1 1
1 2 2 18
2 8 9 243
3 48 75 3240
4 509 934 43239
5 6198 12077 574908
6 80178 159131 7618438
7 1053077 2101575 100803036
8 13890036 27762103 1332343288
9 183339529 366611212 17596479795
10 2419418798 4838564147 232248063316
11 31909900767 63818720716 3063288809012
12 420569653153 841134600018 40374425656248
Code:
d mod M + inv mod M positions
-- ------------ ------------ --------------
0 1 1 1
1 3 3 19
2 11 12 262
3 59 87 3502
4 568 1021 46741
5 6766 13098 621649
6 86944 172229 8240087
7 1140021 2273804 109043123
8 15030057 30035907 1441386411
9 198369586 396647119 19037866206
10 2617788384 5235211266 251285929522
11 34527689151 69053931982 3314574738534
12 455097342304 910188532000 43689000394782