A137457 - OEIS

%I #5 Feb 16 2025 08:33:07

%S 0,1,1,2,1,1,3,1,1,2,1,1,3,1,1,2,1,1,4,1,1,2,1,1,3,1,1,2,1,1,3,1,1,2,

%T 1,1,5,1,1,2,1,1,3,1,1,2,1,1,3,1,1,2,1,1,4,1,1,2,1,1,3,1,1,2,1,1,3,1,

%U 1,2,1,1,5,1,1,2,1,1,3,1,1,2,1,1,3,1,1,2,1,1,4,1,1,2,1,1,3,1,1,2,1,1,3,1,1

%N Consider a row of standard dice as a counter. This sequence enumerates the number of changes (one face rotated over an edge to an adjacent face) from n-1 to n.

%C Most counters 'zero' out at '0' but the dice 'zero' out at '1' which is the initial state. So to increment 1 -> 2 requires 1 move, 2 -> 3 requires 1 move, 3 -> 4 requires 2 moves, 4 -> 5 requires 1 move, 5 -> 6 requires 1 move and 6 -> 0 requires 2 moves.

%C First occurrence of k (A026532): 1, 3, 6, 18, 36, 108, 216, 648, 1296, 3888, ....

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Dice.html">Dice</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dice">Dice</a>.

%t f[n_] := Block[{a = IntegerDigits[n - 1, 6] + 1, b = IntegerDigits[n, 6] + 1, c}, If[Length@b > Length@a, a = Prepend[a, 1]]; c = Transpose[{a, b}] /. {{d_, d_} -> 0, {1, 2} -> 1, {2, 3} -> 1, {3, 4} -> 2, {4, 5} -> 1, {5, 6} -> 1, {6, 1} -> 2}; Plus @@ c]; Array[f, 105]

%Y Cf. A057436, A060852, A026532, A133794.

%K nonn

%O 0,4

%A _Robert G. Wilson v_, Apr 18 2008