A327565 -id:A327565 - cp's OEIS frontend

A327613 Number of transfers of marbles between three sets until the first repetition.

3, 4, 8, 6, 6, 6, 8, 9, 9, 8, 11, 9, 9, 8, 11, 13, 12, 9, 11, 13, 12, 9, 11, 13, 12, 11, 14, 13, 12, 11, 14, 13, 12, 11, 14, 13, 12, 11, 14, 16, 12, 11, 14, 16, 12, 11, 14, 13, 12, 14, 14, 13, 12, 14, 14, 16, 12, 14, 14, 16, 12, 14, 14, 16, 14, 14, 14, 16, 14, 14

Offset: 1

Tristan Cam, Sep 19 2019

nonn

There are initially n marbles in each of the three sets. In the first turn, half of the marbles of set A are transferred to set B, rounding to the upper integer when halving. In the second turn, half of the marbles of set B are transferred to set C, following the same rule. The game goes on back on following the pattern (A to B), (B to C), (C to A) etc. until we reach a distribution already encountered.
a(n) is then the number of steps until the first repetition occurs.
The indexes of the maximal values are 1, 2, 3, 8, 11, 16, 27, 40, 83, 176, 179, 528, 907, 1256, 2379, 3408, ...

			For n = 2, (SetA ; SetB ; SetC):
(2 ; 2 ; 2), ceiling(2/2)=1 marble get transferred from SetA to SetB,
(1 ; 3 ; 2), ceiling(3/2)=2 marbles get transferred from SetB to SetC,
(1 ; 1 ; 4), ceiling(4/2)=2 marbles get transferred from SetC to SetA,
(3 ; 1 ; 2), ceiling(3/2)=2 marbles get transferred from SetA to SetB,
(1 ; 3 ; 2), this is a repetition, it took 4 steps to get there, so a(2) = 4.
For n = 4, (SetA ; SetB ; SetC):
(4 ; 4 ; 4), (2 ; 6 ; 4), (2 ; 3 ; 7), (6 ; 3 ; 3), (3 ; 6 ; 3), (3 ; 3 ; 6), (6 ; 3 ; 3) which is a repetition, so a(4) = 6.

Cf. A327565 (two sets), A327614 (four sets).

A327614 Number of transfers of marbles between four sets until the first repetition.

Original entry on oeis.org

4, 5, 10, 11, 12, 12, 12, 10, 15, 17, 12, 12, 12, 15, 17, 16, 16, 15, 19, 17, 17, 15, 15, 19, 22, 17, 16, 15, 19, 22, 21, 19, 19, 24, 26, 21, 19, 19, 24, 26, 21, 19, 19, 24, 26, 21, 19, 23, 28, 26, 21, 19, 23, 28, 26, 21, 19, 23, 28, 26, 21, 19, 23, 28

Offset: 1

Tristan Cam, Sep 19 2019

nonn

There are initially n marbles in each of the four sets. In the first turn, half of the marbles of set A are transferred to set B, rounding to the upper integer when halving. In the second turn, half of the marbles of set B are transferred to set C, following the same rule. The game goes on back on following the pattern (A to B), (B to C), (C to D), (D to A) etc. until we reach a distribution already encountered.
a(n) is then the number of steps until the first repetition occurs.
The indexes of the maximal values are 1, 2, 3, 4, 5, 9, 10, 19, 25, 34, 35, 49, 105, 194, 330, 334, 480, 1553, 1780, 2834, 2870, 4079, ...

			For n = 2, (SetA ; SetB ; SetC ; SetD):
(2 ; 2 ; 2 ; 2), ceiling(2/2)=1 marble get transferred from SetA to SetB,
(1 ; 3 ; 2 ; 2), ceiling(3/2)=2 marbles get transferred from SetB to SetC,
(1 ; 1 ; 4 ; 2), ceiling(4/2)=2 marbles get transferred from SetC to SetD,
(1 ; 1 ; 2 ; 4), ceiling(4/2)=2 marbles get transferred from SetD to SetA,
(3 ; 1 ; 2 ; 2), ceiling(3/2)=2 marbles get transferred from SetA to SetB,
(1 ; 3 ; 2 ; 2), this is a repetition, it took 5 steps to get there, so a(2) = 5.
For n = 4, (SetA ; SetB ; SetC ; SetD):
(4 ; 4 ; 4 ; 4), (2 ; 6 ; 4 ; 4), (2 ; 3 ; 7 ;4), (2 ; 3 ; 3 ; 8), (6 ; 3 ; 3 ; 4), (3 ; 6 ; 3 ; 4), (3 ; 3 ; 6 ; 4), (3 ; 3 ; 3 ; 7), (7 ; 3 ; 3 ; 3), (3 ; 7 ; 3 ; 3), (3 ; 3 ; 7 ; 3), (3 ; 3 ; 3 ; 7) which is a repetition, so a(4) = 11.

Cf. A327565 (two sets), A327613 (three sets).

cp's OEIS Frontend

A327613 Number of transfers of marbles between three sets until the first repetition.

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