A002491 Smallest number of stones in Tchoukaillon (or Mancala, or Kalahari) solitaire that make use of n-th hole.

1, 2, 4, 6, 10, 12, 18, 22, 30, 34, 42, 48, 58, 60, 78, 82, 102, 108, 118, 132, 150, 154, 174, 192, 210, 214, 240, 258, 274, 282, 322, 330, 360, 372, 402, 418, 442, 454, 498, 510, 540, 570, 612, 622, 648, 672, 718, 732, 780, 802, 840, 870, 918

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

To get n-th term, start with n and successively round up to next multiple of n-1, n-2, ..., 1.

Generated by a sieve: start with [ 1..n ]; keep first number, drop every 2nd, keep first, drop every 3rd, keep first, drop every 4th, etc.

Comment from Don Knuth, Jun 08 2021, added by N. J. A. Sloane, Jun 09 2021: (Start)

I have put together a preliminary exposition of the results of Broline and Loeb (1995), in what's currently called Exercise 11 (on page 7) and its answer (on pages 9 and 10) of the Bipartite Matching document.

Unfortunately, I believe this paper erred when it claimed to have proved the conjecture of Erdős and Jabotinsky about the sharpness of approximation to n^2/pi. When they computed the sum of I_M in the proof of Theorem 9, they expressed it as f(M-1)^2/4M + O(f(M-1)). That's correct; but the error gets quite large, and when summed gives O(n^(3/2)), not O(n).

By summing over only part of the range, and using another estimate in the rest, I believe one can get an overall error bound O(n^(4/3)), thus matching the result of Erdős and Jabotinsky but not improving on it. (End)

Examples

			To get 10th term: 10->18->24->28->30->30->32->33->34->34.
		

References

• Y. David, On a sequence generated by a sieving process, Riveon Lematematika, 11 (1957), 26-31.
• S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.4.7.
• V. Gautheron, Chapter 3.II.5: La Tchouka, in Wari et Solo: le Jeu de calculs africain (Les Distracts), Edited by A. Deledicq and A. Popova, CEDIC, Paris, 1977, 180-187.
• D. E. Knuth, Bipartite Matching, The Art of Computer Programming, Vol. 4, Pre-fascicle 14A, June 8, 2021, http://cs.stanford.edu/~knuth/fasc14a.ps.gz. See Sect. 7.5.1, Exercise 11.
• 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).

Links

• Kerry Mitchell, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• D. Betten, Kalahari and the Sequence "Sloane No. 377", Annals Discrete Math., 37, 51-58, 1988.
• 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; arXiv:math/9502225 [math.CO], 1995.
• K. S. Brown, Rounding Up To PI
• Y. David, On a sequence generated by a sieving process, Riveon Lematematika, 11 (1957), 26-31. [Annotated scan of pages 31 and 27 only]
• Mark Dukes, Sequences of integer pairs related to the game of Tchoukaillon solitaire, University College Dublin (Ireland, 2020).
• Mark Dukes, Fagan's Construction, Strange Roots, and Tchoukaillon Solitaire, Journal of Integer Sequences, Vol. 24 (2021), Article 21.7.1.
• Mark Dukes, Fagan's Construction, Strange Roots, and Tchoukaillon Solitaire, arXiv:2202.02381 [math.NT], 2022.
• P. Erdős and E. Jabotinsky, On a sequence of integers generated by a sieving process (Part I), Indagationes Math., 20, 115-128, 1958.
• P. Erdős and E. Jabotinsky, On a sequence of integers generated by a sieving process (Part II), Indagationes Math., 20, 115-128, 1958.
• B. Gourevitch, The World of Pi
• Nick Hobson, Python program for this sequence
• Brant Jones, Laura Taalman and Anthony Tongen, Solitaire Mancala Games and the Chinese Remainder Theorem, Amer. Math. Mnthly, 120 (2013), 706-724.
• N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
• Eric Weisstein's World of Mathematics, Pi.
• Eric Weisstein's World of Mathematics, Pi Formulas.
• Index entries for sequences generated by sieves

Crossrefs

Cf. A000012, A000960, A028920, A028931, A028932, A028933, A112557, A112558, A113742, A113743, A113744, A113745, A113746, A113747, A113748, A113749.

Cf. also A104738.

A row of the array in A344009.

Programs

Haskell

a002491 n = a002491_list !! (n-1)
a002491_list = sieve 1 [1..] where
   sieve k (x:xs) = x : sieve (k+1) (mancala xs) where
      mancala xs = us ++ mancala vs where (us,v:vs) = splitAt k xs
-- Reinhard Zumkeller, Oct 31 2012

Maple

# A002491
# program due to B. Gourevitch
a := proc(n)
local x,f,i,y;
  x := n; f := n;
  for i from x by -1 to 2 do
     y := i-1;
     while y < f do
       y := y+i-1
     od;
  f := y
  od
end:
seq(a(n), n = 2 .. 53);

Mathematica

f[n_] := Fold[ #2*Ceiling[ #1/#2 + 0] &, n, Reverse@Range[n - 1]]; Array[f, 56] (* Robert G. Wilson v, Nov 05 2005 *)
del[list_, k_] := Delete[list, Table[{i}, {i, k, Length[list], k}]]; a[n_] := Last@NestWhile[{#[[1]] + 1, del[Rest@#[[2]], #[[1]] + 1], Append[#[[3]], First@#[[2]]]} &, {1,Range[n], {}}, #[[2]] =!= {} &]; a[1000] (* Birkas Gyorgy, Feb 26 2011 *)
Table[1 + First@FixedPoint[{Floor[#[[1]]*(#[[2]] + 1/2)/#[[2]]], #[[2]] + 1} &, {n, 1}, SameTest -> (#1[[1]] == #2[[1]] &)], {n, 0, 30}] (* Birkas Gyorgy, Mar 07 2011 *)
f[n_]:=Block[{x,p},For[x=p=1, p<=n, p++, x=Ceiling[(n+2-p)x/(n+1-p)]];x] (* Don Knuth, May 27 2021 *)

PARI

a(n)=forstep(k=n-1,2,-1, n=((n-1)\k+1)*k); n \\ Charles R Greathouse IV, Mar 29 2016

Formula

Equals A007952(n) + 1 or equally A108696(n) - 1.

A130747(a(n)) = 1. - Reinhard Zumkeller, Jun 23 2009

a(n+1) = 1 + [..[[[[n*3/2]5/4]7/6]9/8]...(2k+1)/2k]...]. - Birkas Gyorgy, Mar 07 2011

Limit_{n -> oo} n^2/a(n) = Pi (see Brown). - Peter Bala, Mar 12 2014

It appears that a(n)/2 = A104738(n-1). - Don Knuth, May 27 2021