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Winning positions in the u-pile of the 4-Wythoff game with i=1.
(Formerly M0943 N0354)
+10
5
0, 1, 2, 4, 5, 6, 7, 8, 10, 11, 12, 13, 15, 16, 17, 18, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 55, 57, 58, 59, 60, 62, 63, 64, 65, 67, 68, 69, 70, 72, 73, 74, 75, 76, 78, 79, 80, 81, 83
COMMENTS
See Connell (1959) for further information.
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(n) = floor( (n+1/4)*(sqrt(5)-1) ). - R. J. Mathar, Feb 14 2011
MATHEMATICA
Table[Floor[(n + 1/4)*(Sqrt[5] - 1)], {n, 0, 100}] (* T. D. Noe, Aug 17 2012 *)
a(n) = floor((n+2/3)*(5+sqrt(13))/2); v-pile positions in the 3-Wythoff game.
(Formerly M1735 N0687)
+10
4
2, 7, 11, 15, 20, 24, 28, 32, 37, 41, 45, 50, 54, 58, 63, 67, 71, 76, 80, 84, 88, 93, 97, 101, 106, 110, 114, 119, 123, 127, 131, 136, 140, 144, 149, 153, 157, 162, 166, 170, 174, 179, 183, 187, 192, 196, 200, 205, 209, 213, 218, 222, 226, 230, 235, 239, 243, 248
COMMENTS
3-Wythoff game, i=2, the v-pile positions in the Connell terminology. - R. J. Mathar, Feb 14 2011
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(n) = floor((n+2/3)*(5+sqrt(13))/2). - R. J. Mathar, Feb 14 2011
MATHEMATICA
Table[Floor[(n + 2/3)*(5 + Sqrt[13])/2], {n, 0, 100}] (* T. D. Noe, Aug 17 2012 *)
u-pile numbers for the 3-Wythoff game with i=2.
(Formerly M0541)
+10
3
0, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 24, 25, 26, 28, 29, 30, 32, 33, 34, 36, 37, 38, 39, 41, 42, 43, 45, 46, 47, 49, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 63, 64, 66, 67, 68, 69, 71, 72, 73, 75, 76, 77, 79, 80, 81, 82, 84, 85, 86, 88
COMMENTS
See Connell (1959) for further information.
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(n) = floor( (n+2/3)*(sqrt(13)-1)/2 ). - R. J. Mathar, Feb 14 2011
MATHEMATICA
Table[Floor[(n + 2/3)*(Sqrt[13] - 1)/2], {n, 0, 100}] (* T. D. Noe, Aug 17 2012 *)
u-pile positions in the 3-Wythoff game with i=1.
(Formerly M2302 N0908)
+10
2
0, 1, 3, 4, 5, 6, 8, 9, 10, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 30, 31, 33, 34, 35, 36, 38, 39, 40, 42, 43, 44, 46, 47, 48, 49, 51, 52, 53, 55, 56, 57, 59, 60, 61, 62, 64, 65, 66, 68, 69, 70, 72, 73, 74, 75, 77, 78, 79, 81, 82, 83, 85, 86, 87
COMMENTS
See Connell (1959) for further information.
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
FORMULA
a(n) = floor((n+1/3)*(sqrt(13)-1)/2). - R. J. Mathar, Feb 14 2011
MATHEMATICA
Table[Floor[(n + 1/3)*(Sqrt[13] - 1)/2], {n, 0, 100}] (* T. D. Noe, Aug 17 2012 *)
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