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a(n) is the number of ways of making n moves in English Peg Solitaire.
+10
2
1, 4, 12, 60, 400, 2960, 24600, 221072, 2076744, 20123080, 197757768, 1937125160, 18687793880, 175793675328, 1594744777464, 13794351556920, 112576101214496, 857945953884624, 6037935953538456, 38729529837059648, 222984258240522544, 1133096911619304064, 4985812137371331624
OFFSET
0,2
COMMENTS
This sequence has 32 terms in total.
EXAMPLE
Given the positions marked thus:
a b c
d e f
g h i j k l m
n o p q r s t
u v w x y z 0
1 2 3
4 5 6
there are 12 ways to make two moves, viz.,
(1) e jumps over j, then h jumps over i;
(2) e jumps over j, then x jumps over q;
(3) e jumps over j, then l jumps over k;
(4) o jumps over p, then d jumps over i;
(5) o jumps over p, then 1 jumps over w;
(6) o jumps over p, then r jumps over q;
(7) 2 jumps over x, then j jumps over q;
(8) 2 jumps over x, then v jumps over w;
(9) 2 jumps over x, then z jumps over y;
(10) s jumps over r, then f jumps over k;
(11) s jumps over r, then p jumps over q;
(12) s jumps over r, then 3 jumps over y.
CROSSREFS
KEYWORD
nonn,fini
AUTHOR
Douglas Boffey, Jan 28 2022
STATUS
approved
a(n) is the number of boards in English Peg Solitaire, reached after n moves, for which no more moves are possible.
+10
1
0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 8, 4, 0, 28, 66, 40, 176, 302, 816, 2675, 6890, 15122, 25846, 33891, 33051, 23722, 13142, 4987, 1295, 254, 28, 5
OFFSET
0,7
CROSSREFS
KEYWORD
nonn,fini,full
AUTHOR
Douglas Boffey, Jan 28 2022
STATUS
approved
a(n) is the number of boards in English Peg Solitaire after n moves from which the winning position can be reached.
+10
1
1, 4, 12, 60, 292, 1292, 5012, 16628, 49236, 127964, 285740, 546308, 902056, 1298248, 1639652, 1841556, 1841556, 1639652, 1298248, 902056, 546308, 285740, 127964, 49236, 16628, 5012, 1292, 292, 60, 12, 4, 1
OFFSET
0,2
COMMENTS
After the first 16 terms the next terms are the first 16 terms in decreasing order. - Omar E. Pol, Feb 06 2022
CROSSREFS
Cf. A335656.
KEYWORD
nonn,fini,full
AUTHOR
Douglas Boffey, Feb 06 2022
STATUS
approved
Number of distinct board states reachable in n jumps in European Peg Solitaire.
+10
1
1, 4, 17, 92, 495, 2475, 11771, 52226, 212527, 789228, 2640323, 7870055, 20730606, 47916748, 96715832, 170154214, 260956703, 349541944, 410294786, 423631649, 385887175, 310724581, 221398196, 139580751, 77748102, 38162987, 16445627, 6178002, 2007607, 559163, 131269, 25378, 4012, 481, 36, 4
OFFSET
0,2
EXAMPLE
The beginning state is missing the peg just above the center, as an initial state with the center peg removed does not yield any valid solutions where 1 peg is remaining.
* * *
* * * * *
* * * O * * *
* * * * * * *
* * * * * * *
* * * * *
* * *
The next move yields the next 4 states:
* * * * * * * O * * * *
* * * * * * * * * * * * O * * * * * * *
* O O * * * * * * * * * * * * * * * * * * * * * * O O *
* * * * * * * * * * O * * * * * * * * * * * * * * * * *
* * * * * * * * * * O * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * *
CROSSREFS
KEYWORD
nonn,fini,full
AUTHOR
Sander G. Huisman, Jun 27 2022
STATUS
approved

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