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A002491
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Smallest number of stones in Tchoukaillon (or Mancala, or Kalahari) solitaire that make use of n-th hole.
(Formerly M1009 N0377)
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25
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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
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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A130747(a(n)) = 1. - Reinhard Zumkeller, Jun 23 2009
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REFERENCES
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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.
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).
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LINKS
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Kerry Mitchell, Table of n, a(n) for n = 1..10000, extending the list submitted by T.D. Noe.
D. Betten, Kalahari and the Sequence "Sloane No. 377", Annals Discrete Math., 37, 51-58, 1988.
K. S. Brown, Rounding Up To PI
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.
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).
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
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FORMULA
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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.
Equals A007952(n)+1 or equally A108696(n)-1.
a(n+1) = 1 + [..[[[[n*3/2]5/4]7/6]9/8]...(2k+1)/2k]...] (Birkas Gyorgy, Mar 07 2011)
n^2/a(n) -> Pi as n -> infinity (see Brown). - Peter Bala, Mar 12 2014
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EXAMPLE
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To get 10th term: 10->18->24->28->30->30->32->33->34->34.
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MAPLE
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# 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);
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MATHEMATICA
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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 *)
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PROG
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(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
(PARI) a(n)=forstep(k=n-1, 2, -1, n=((n-1)\k+1)*k); n \\ Charles R Greathouse IV, Mar 29 2016
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CROSSREFS
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Cf. A000012, A000960, A028920, A028931, A028932, A028933, A112557, A112558, A113742, A113743, A113744, A113745, A113746, A113747, A113748, A113749.
Sequence in context: A002088 A092249 A019332 * A045958 A076067 A316460
Adjacent sequences: A002488 A002489 A002490 * A002492 A002493 A002494
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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