Search: a112737 -id:a112737
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A335656
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Number of distinct board states reachable in n jumps, in English Peg Solitaire.
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+10
4
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1, 4, 12, 60, 296, 1338, 5648, 21842, 77559, 249690, 717788, 1834379, 4138302, 8171208, 14020166, 20773236, 26482824, 28994876, 27286330, 22106348, 15425572, 9274496, 4792664, 2120101, 800152, 255544, 68236, 14727, 2529, 334, 32, 5
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OFFSET
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0,2
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LINKS
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EXAMPLE
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Example: for n=1 the four states are:
*** *** *** ***
*.* *** *** ***
***.*** ******* ******* *******
******* ****..* ******* *..****
******* ******* ***.*** *******
*** *** *.* ***
*** *** *** ***
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CROSSREFS
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Identifying positions that are related by a symmetry of the board gives A112737.
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KEYWORD
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fini,full,nonn
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AUTHOR
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STATUS
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approved
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A355295
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Number of distinct board states reachable in n jumps in European Peg Solitaire.
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+10
1
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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
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OFFSET
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0,2
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LINKS
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EXAMPLE
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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 * * * * * * * * * * * * * * * * *
* * * * * * * * * * * * * * * * * * * *
* * * * * * * * * * * *
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CROSSREFS
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KEYWORD
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nonn,fini,full
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AUTHOR
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STATUS
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approved
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A112738
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On the standard 33-hole cross-shaped peg solitaire board, the number of distinct board positions after n jumps that can still be reduced to one peg at the center (starting with the center vacant).
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+10
0
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1, 1, 2, 8, 38, 164, 635, 2089, 6174, 16020, 35749, 68326, 112788, 162319, 204992, 230230, 230230, 204992, 162319, 112788, 68326, 35749, 16020, 6174, 2089, 635, 164, 38, 8, 2, 1, 1, 0
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OFFSET
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0,3
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COMMENTS
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The reason the sequence is palindromic is because playing the game backward is the same as playing it forward, with the notions of "hole" and "peg" interchanged.
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LINKS
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FORMULA
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Satisfies a(n)=a(31-n) for 0<=n<=31 (sequence is a palindrome).
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EXAMPLE
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There are four possible first jumps, but they all lead to the same board position (rotationally equivalent), thus a(1)=1.
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CROSSREFS
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KEYWORD
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full,nonn,fini
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AUTHOR
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George Bell (gibell(AT)comcast.net), Sep 16 2005
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STATUS
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approved
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