A028931 Strings giving winning positions in Tchoukaillon (or Mancala) solitaire.

0, 1, 20, 21, 310, 311, 4200, 4201, 4220, 4221, 53110, 53111, 642000, 642001, 642020, 642021, 642310, 642311, 7531200, 7531201, 7531220, 7531221, 86420110, 86420111, 86424000, 86424001, 86424020, 86424021, 86424310, 86424311

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• nice
• easy
• base

Comments

a(n) gives string listing winning position for n stones.

Sum of numbers in a(n) is equal to n.

Using the chromatic (tet-12) scale, (if C=0, 12, 24...) all integers correspond to pitches C, C#, G#, A, Bb, B, a recursive pattern that evenly bisects the octave and avoids the tritone (IC6). - Nik Bizzell-Browning, Apr 27 2019

Examples

			For example, a(10) = 53111; rightmost 0 is in position 6, so get 653111 -> a(11) = 642000.
		

Links

• Nik Bizzell-Browning, Tchoukaillon solitaire sequence, Chromatic pitch table and example.
• Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 142.
• D. M. Broline and _Daniel E. Loeb_, The combinatorics of Mancala-Type games: Ayo, Tchoukaillon and 1/Pi, J. Undergrad. Math. Applic., vol. 16 (1995), pp. 21-36.

Crossrefs

Cf. A002491, A007952, A028920, A028932, A028933.

Programs

Mathematica

a[1] = {0};a[n_]:=a[n]=Module[{gs=a[n-1],len=Length[a[n-1]],pos},If[FreeQ[gs,0],Prepend[gs-1,1+len],pos=FirstPosition[Reverse[gs],0][[1]];(gs[[;;len-pos]])~Join~{pos}~Join~If[pos==1,{},(gs[[len-pos+2;;]]-1)]]];StringJoin/@Map[ToString, Array[a, 30], {2}] (* Shenghui Yang, Jun 06 2025 *)

Formula

To get the next term, if rightmost 0 is in position i, replace it by i and subtract 1 from all earlier entries.

Extensions

More terms from Erich Friedman