A002491 -id:A002491 - cp's OEIS frontend

A113750 Consider the generalized Mancala solitaire (A002491); to get n-th term, start with n and successively round up to next k multiples of n-1, n-2, ..., 1, for n>=1. Now construct the array, t, such that t(n,k) is the n-th and successively rounding up to the next k multiples. This sequence is the determinant of that array.

2, 2, -4, 0, -32, -256, -512, -5632, -180736, -135168, -61440, 5529600, -1554161664, 17735712768, 351786369024, -79390588010496, -1755801711804416, -30318369806745600, -4162409018839531520, 528913148312239996928

Offset: 2

Paul D. Hanna and Robert G. Wilson v, Nov 05 2005

sign

f[n_, k_] := Fold[ #2*Ceiling[ #1/#2 + k] &, n, Reverse@Range[n - 1]]; Table[Det[Table[f[i, j], {i, 2, n}, {j, 0, n - 2}]], {n, 2, 21}]

A000960 Flavius Josephus's sieve: Start with the natural numbers; at the k-th sieving step, remove every (k+1)-st term of the sequence remaining after the (k-1)-st sieving step; iterate.

Original entry on oeis.org

1, 3, 7, 13, 19, 27, 39, 49, 63, 79, 91, 109, 133, 147, 181, 207, 223, 253, 289, 307, 349, 387, 399, 459, 481, 529, 567, 613, 649, 709, 763, 807, 843, 927, 949, 1009, 1093, 1111, 1189, 1261, 1321, 1359, 1471, 1483, 1579, 1693, 1719, 1807, 1899, 1933, 2023

Offset: 1

N. J. A. Sloane

a(n) is never divisible by 2 or 5. - Thomas Anton, Nov 01 2018

			Start with
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ... (A000027) and delete every second term, giving
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 ... (A005408) and delete every 3rd term, giving
1 3 7 9 13 15 19 21 25 27 ... (A056530) and delete every 4th term, giving
1 3 7 13 15 19 25 27 ... (A056531) and delete every 5th term, giving
.... Continue forever and what's left is the sequence.
(The array formed by these rows is A278492.)
For n = 5, 5^2 = 25, go down to a multiple of 4 giving 24, then to a multiple of 3 = 21, then to a multiple of 2 = 20, then to a multiple of 1 = 19, so a(5) = 19.

V. Brun, Un procédé qui ressemble au crible d'Ératosthène, Analele Stiintifice Univ. "Al. I. Cuza", Iasi, Romania, Sect. Ia Matematica, 1965, vol. 11B, pp. 47-53.
Problems 107, 116, Nord. Mat. Tidskr. 5 (1957), 114-116.
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).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. E. Andersson, Das Flaviussche Sieb, Acta Arith., 85 (1998), 301-307.
L. Carlitz and Chr. U. Jansen, Solution to Problem 115 and 116, Nord. Mat. Tidskr. 5 (1957), 159-161.
V. Gardiner, R.Lazarus, N. Metropolis and S. Ulam, On certain sequences of integers defined by sieves, Math. Mag., 29 (1955), 117-119.
H. Killingbergtro, Solution to Problem 107, Nord. Mat. Tidskr. 5 (1957), 203-205.
D. Wilson et al., Interesting sequence, SeqFan list, Nov. 2016
Index entries for sequences related to the Josephus Problem
Index entries for sequences generated by sieves

Cf. A056526, A056530, A056531, A100002, A190732, A003881.
Cf. A000012, A002491, A000959, A003309, A099259, A112557, A112558, A113742, A113743, A113744, A113745, A113746, A113747, A113748; A113749, A278484.
Cf. A119446 for triangle whose leading diagonal is A119447 and this sequence gives all possible values for A119447 (except A119447 cannot equal 1 because prime(n)/n is never 1).
Cf. A100617 (a left inverse), A100618.
Cf. A278169 (characteristic function).
Main diagonal of A278492, leftmost column of A278505, positions of zeros in A278528 & A278529.

a000960 n = a000960_list !! (n-1)
a000960_list = sieve 1 [1..] where
   sieve k (x:xs) = x : sieve (k+1) (flavius xs) where
      flavius xs = us ++ flavius vs where (u:us,vs) = splitAt (k+1) xs
-- Reinhard Zumkeller, Oct 31 2012

S[1]:={seq(i,i=1..2100)}: for n from 2 to 2100 do S[n]:=S[n-1] minus {seq(S[n-1][n*i],i=1..nops(S[n-1])/n)} od: A:=S[2100]; # Emeric Deutsch, Nov 17 2004

del[lst_, k_] := lst[[Select[Range[Length[lst]], Mod[ #, k] != 0 &]]]; For[k = 2; s = Range[2100], k <= Length[s], k++, s = del[s, k]]; s
f[n_] := Fold[ #2*Ceiling[ #1/#2 + 1] &, n, Reverse@Range[n - 1]]; Array[f, 60] (* Robert G. Wilson v, Nov 05 2005 *)

a(n)=local(A=n,D);for(i=1,n-1,D=n-i;A=D*ceil(A/D+1));return(A) \\ Paul D. Hanna, Oct 10 2005

def flavius(n):
    L = list(range(1,n+1));j=2
    while j <= len(L):
        L = [L[i] for i in range(len(L)) if (i+1)%j]
        j+=1
    return L
flavius(100)
# Robert FERREOL, Nov 08 2015

Let F(n) = number of terms <= n. Andersson, improving results of Brun, shows that F(n) = 2 sqrt(n/Pi) + O(n^(1/6)). Hence a(n) grows like Pi*n^2 / 4.
To get n-th term, start with n and successively round up to next 2 multiples of n-1, n-2, ..., 1 (compare to Mancala sequence A002491). E.g.: to get 11th term: 11->30->45->56->63->72->80->84->87->90->91; i.e., start with 11, successively round up to next 2 multiples of 10, 9, .., 1. - Paul D. Hanna, Oct 10 2005
As in Paul D. Hanna's formula, start with n^2 and successively move down to the highest multiple of n-1, n-2, etc., smaller than your current number: 121 120 117 112 105 102 100 96 93 92 91, so a(11) = 91, from moving down to multiples of 10, 9, ..., 1. - Joshua Zucker, May 20 2006
Or, similarly for n = 5, begin with 25, down to a multiple of 4 = 24, down to a multiple of 3 = 21, then to a multiple of 2 = 20 and finally to a multiple of 1 = 19, so a(5) = 19. - Joshua Zucker, May 20 2006
This formula arises in A119446; the leading term of row k of that triangle = a(prime(k)/k) from this sequence. - Joshua Zucker, May 20 2006
a(n) = 2*A073359(n-1) + 1, cf. link to posts on the SeqFan list. - M. F. Hasler, Nov 23 2016
a(n) = 1 + A278484(n-1). - Antti Karttunen, Nov 23 2016, after David W. Wilson's posting on SeqFan list Nov 22 2016

More terms and better description from Henry Bottomley, Jun 16 2000
Entry revised by N. J. A. Sloane, Nov 13 2004
More terms from Paul D. Hanna, Oct 10 2005

A007952 Generated by a sieve: keep first number, drop every 2nd, keep first, drop every 3rd, keep first, drop every 4th, etc.

Original entry on oeis.org

0, 1, 3, 5, 9, 11, 17, 21, 29, 33, 41, 47, 57, 59, 77, 81, 101, 107, 117, 131, 149, 153, 173, 191, 209, 213, 239, 257, 273, 281, 321, 329, 359, 371, 401, 417, 441, 453, 497, 509, 539, 569, 611, 621, 647, 671, 717, 731, 779, 801, 839, 869, 917, 929, 989, 1001, 1053, 1067

Offset: 0

N. J. A. Sloane, R. Muller

Also called the sieve of Tchoukaillon (or Mancala, or Kalahari).
If k+1 occurs at rank i for the first time, then i is given by the program: i = 0: for j = k to 1 step -1: i = 1 + i + int ( i / j ): next: - Claude Lenormand (claude.lenormand(AT)free.fr), Jan 15 2001
A082447(n+1) = (number of terms <= n); see A141262 for primes. - Reinhard Zumkeller, Jun 21 2008

Y. David, On a sequence generated by a sieving process, Riveon Lematematika, 11 (1957), 26-31.
M. Le, On the Smarandache n-ary Sieve, Smarandache Notions Journal, Vol. 10, No. 1-2-3, 1999, 146-147.

L. K. Mitchell, Table of n, a(n) for n = 0..7549
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, arXiv:math/9502225 [math.CO], 1995; J. Undergrad. Math. Applic., vol. 16 (1995), pp. 21-36.
P. Erdős and E. Jabotinsky, On a sequence of integers ..., Indagationes Math., 20, 115-128, 1958. part I part II
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
F. Smarandache, Only Problems, Not Solutions!
Index entries for sequences generated by sieves

Cf. A002491, A028920, A028931, A028932, A028933, A108696, A140060, A141271, A141272.

a007952 n = a007952_list !! n
a007952_list = f 1 [0..] where
   f k (x:xs) = x : f (k + 1) (g xs) where
     g ws = us ++ (g vs) where (us, _:vs) = splitAt k ws
-- Reinhard Zumkeller, Jan 19 2014

f[n_] := Fold[#2*Floor[#1/#2 + 1] &, n, Reverse@ Range[n - 1]]; Array[f, 55] (* From David Wilson *)

a(n) = my(ret=0); forstep(k=n,1,-1, ret++; ret+=(-ret)%k); ret; \\ Kevin Ryde, Sep 30 2022

Equals A002491(n) - 1. Equals A108696 - 2.

Corrected and extended by David W. Wilson

A028920 Pit harvesting sequence for winning solitaire Tchoukaillon (or Mancala).

Original entry on oeis.org

1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 1, 6, 1, 2, 1, 3, 1, 7, 1, 2, 1, 8, 1, 4, 1, 2, 1, 3, 1, 9, 1, 2, 1, 10, 1, 5, 1, 2, 1, 3, 1, 11, 1, 2, 1, 4, 1, 12, 1, 2, 1, 3, 1, 6, 1, 2, 1, 13, 1, 14, 1, 2, 1, 3, 1, 4, 1, 2, 1, 5, 1, 7, 1, 2, 1, 3, 1, 15, 1, 2, 1, 16, 1, 4, 1, 2, 1, 3, 1, 8, 1, 2, 1, 6, 1, 5, 1, 2, 1, 3, 1, 17, 1

Offset: 0

David W. Wilson

nonn

From Benoit Cloitre, Mar 09 2007: (Start)
The sequence can be constructed as follows using parentheses (NP means "term not in parentheses"):
Start from the positive integers:
1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,...
Step 1: put the least NP "1" in parentheses and every 2 terms giving:
(1),2,(3),4,(5),6,(7),8,(9),10,(11),12,(13),14,(15),16,(17),18,(19),...
Step 2: put the least NP "2" in 2 parentheses and every 3 NP giving:
(1),((2)),(3),4,(5),6,(7),((8)),(9),10,(11),12,(13),((14)),(15),16,(17),...
so that between 2 consecutives ((x)) there are 2 NP.
Step 3: put the least NP "4" in 3 parentheses and every 4 NP giving:
(1),((2)),(3),(((4))),(5),6,(7),((8)),(9),10,(11),12,(13),((14)),(15),(((16))),...
so that between 2 consecutives (((x))) there are 3 NP.
Step 4: put the least NP "6" in 4 parentheses and every 5 NP giving:
(1),((2)),(3),(((4))),(5),((((6)))),(7),((8)),(9),10,(11),12,(13),((14)),(15),(((16))),...
so that between 2 consecutives ((((x)))) there are 4 NP.
Iterating the process indefinitely yields:
(1),((2)),(3),(((4))),(5),((((6)))),(7),((8)),(9),(((((10))))),(11),...
Count the parentheses:
1,2,1,3,1,4,1,2,1,5,1,... - this is the sequence.
(End)
From Benoit Cloitre, Jul 26 2007: (Start)
A simpler way to construct the sequence: start from
1,,1,,1,,1,,1,,1,,1,_,1,... where 1's are spaced by one hole;
fill first hole with 2 and leave 2 holes between two 2's giving
1,2,1,,1,,1,2,1,,1,,1,2,1,...;
fill new first hole with 3 and leave 3 holes between two 3's giving
1,2,1,3,1,,1,2,1,,1,_,1,2,1,3...;
iterating the process indefinitely yields the sequence.
(End)
Ordinal transform of A130747. - Benoit Cloitre, Aug 03 2007
Although this sequence and A130747 are not fractal sequences (according to Kimberling's definition), we say they are "mutual fractal sequences" since the ordinal transform of one gives the other. - Benoit Cloitre, Aug 03 2007
The smallest n with a(n) = k is circa k^2/Pi.
The element n >= 0 occurs in this sequence with limiting density 1/(n*(n+1)).

L. K. Mitchell, Table of n, a(n) for n=0..3280
Franklin T. Adams-Watters, Doubly Fractal Sequences and ordinal transform
D. M. Broline and Daniel E. Loeb, The combinatorics of Mancala-Type games: Ayo, Tchoukaillon and 1/Pi, arXiv:math/9502225 [math.CO], 1995; J. Undergrad. Math. Applic., vol. 16 (1995), pp. 21-36.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
Index entries for sequences generated by sieves

Cf. A002491, A007952, A028931, A028932, A028933, A130747.

n = 15; Fold[If[Length@Position[#1, 0] > 0, ReplacePart[#1,First /@ Partition[Position[#1, 0], #2 + 1, #2 + 1, {1, 1}] -> #2], #1] &,  Flatten@Array[{1, 0} &, n], Range[2, 2 n]] (* Birkas Gyorgy, Feb 26 2011 *)

a(n) = {ok = 0; m = 1; while (!ok, if ((n%(m+1) == 0), ok = 1, n = n*m\(m+1); m++);); m;} \\ Michel Marcus, Dec 06 2015

a(2n+1) = 1 + A104706(n+1), a(2n) = 1. - Benoit Cloitre, Mar 09 2007

The sieve of A007952 processes n in the a(n)-th pass. a(A007952(n)) = n+1.

Additional comments from David W. Wilson, Feb 25 2010

A112564 Square array, read by ascending antidiagonals, where each row is a generalized Flavius Josephus sieve (A000960).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 4, 1, 1, 4, 7, 6, 1, 1, 5, 13, 13, 10, 1, 1, 6, 21, 28, 19, 12, 1, 1, 7, 31, 61, 61, 27, 18, 1, 1, 8, 43, 96, 125, 88, 39, 22, 1, 1, 9, 57, 169, 241, 261, 133, 49, 30, 1, 1, 10, 73, 232, 505, 546, 421, 208, 63, 34, 1, 1, 11, 91, 361, 785, 1051, 1171, 605, 313

Offset: 0

Paul D. Hanna, Oct 14 2005

			Table begins:
  1,  1,  1,   1,    1,    1,     1,     1,      1,     1, ...
  1,  2,  4,   6,   10,   12,    18,    22,     30,    34, ...
  1,  3,  7,  13,   19,   27,    39,    49,     63,    79, ...
  1,  4, 13,  28,   61,   88,   133,   208,    313,   364, ...
  1,  5, 21,  61,  125,  261,   421,   605,   1101,  1681, ...
  1,  6, 31,  96,  241,  546,  1171,  1776,   2761,  5046, ...
  1,  7, 43, 169,  505, 1051,  2527,  5083,  7729,  11635, ...
  1,  8, 57, 232,  785, 1800,  5041, 11096, 22737,  34504, ...
  1,  9, 73, 361, 1153, 3961,  8281, 20161, 43633,  95049, ...
  1, 10, 91, 460, 1981, 5950, 13951, 38080, 91081, 186130, ...
  ...

Index entries for sequences related to the Josephus Problem

Cf. A002491 (row 1), A000960 (row 2), A112560 (row 3), A112561 (row 4), A112562 (row 5), A112563 (row 6), A112565 (main diagonal), A112568 (2nd diagonal), A112569 (antidiagonal sums).

{T(n,k)=local(A=k,B=0,C=0);if(n==0||k==0,1, until(A==B,C=C+1;if(C%n==0,C=C+1);B=A;A=floor(A*(C+1)/C));1+A)}

A112565 Main diagonal of square table A112564 of generalized Flavius Josephus sieves.

Original entry on oeis.org

1, 2, 7, 28, 125, 546, 2527, 11096, 43633, 186130, 809831, 3423432, 14022373, 58574894, 250708291, 1038612976, 4353755777, 18333324162, 74663736859, 311293807040, 1286700247561, 4917768055222, 20458840039199, 83985256000824

Offset: 0

Paul D. Hanna, Oct 14 2005

nonn

Appears to be a self-convolution of an integer sequence (A112567).

Index entries for sequences related to the Josephus Problem

Cf. A112564, A112566, A112567, A112568.

{a(n)=local(A=n,B=0,C=0);if(n==0,1, until(A==B,C=C+1;if(C%n==0,C=C+1);B=A;A=floor(A*(C+1)/C));1+A)}

A002491(n) = local(a, b); a = n; b = n - 1; while (b > 1, a = b*ceil(a/b); b--); a;
T(n, k) = local(A = k, C = 1, q, d, x); if (n*k == 0, return(1)); if (n == 1, return(A002491(k + 1))); while (q = A\C, d = A%C; x = d\q + 1; A += x*(n - 1)*(A\C); C += x*n); 1 + A; \\ David Wasserman, Jun 25 2009

a(n) = 1 + n*A112566(n) for n >= 0.

More terms from David Wasserman, Jun 25 2009

A028933 Table of winning positions in Tchoukaillon (or Mancala) solitaire.

Original entry on oeis.org

0, 1, 0, 2, 1, 2, 0, 1, 3, 1, 1, 3, 0, 0, 2, 4, 1, 0, 2, 4, 0, 2, 2, 4, 1, 2, 2, 4, 0, 1, 1, 3, 5, 1, 1, 1, 3, 5, 0, 0, 0, 2, 4, 6, 1, 0, 0, 2, 4, 6, 0, 2, 0, 2, 4, 6, 1, 2, 0, 2, 4, 6, 0, 1, 3, 2, 4, 6, 1, 1, 3, 2, 4, 6, 0, 0, 2, 1, 3, 5, 7, 1, 0, 2, 1, 3, 5, 7

Offset: 0

N. J. A. Sloane

Table read by rows where b(n,i) = the number of counters in the i-th position from the store of the unique winning Tchoukaillon board having n total counters.

			   The rows of b(n,i) begin
   n\i 1 2 3 4 5 6
   1   1
   2   0 2
   3   1 2
   4   0 1 3
   5   1 1 3
   6   0 0 2 4
   7   1 0 2 4
   8   0 2 2 4
   9   1 2 2 4
   10  0 1 1 3 5
   11  1 1 1 3 5
   12  0 0 0 2 4 6
   13  1 0 0 2 4 6
   14  0 2 0 2 4 6
   15  1 2 0 2 4 6
   16  0 1 3 2 4 6
   17  1 1 3 2 4 6

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.
Brant Jones, Laura Taalman and Anthony Tongen, Solitaire Mancala Games and the Chinese Remainder Theorem, Amer. Math. Monthly, 120 (2013), 706-724.

Cf. A002491, A028920, A007952, A028931, A028932, A028933.

s[list_] := Module[{x = Append[list, 0], i = 1}, While[x[[i]] =!= 0, x[[i]] = x[[i]] - 1; i = i + 1]; x[[i]] = i; If[Last@x == 0, Most[x], x]]; Prepend[Flatten@NestList[s, {}, 20],0] (* Birkas Gyorgy, Feb 26 2011 *)

Let p(n) be the minimum j such that b(n,j) = 0.  (This is A028920.)
Directly from the rules of Tchoukaillon, we find b(n+1,i) = (b(n,i) - 1 for 1 <= i < p(n), i for i = p(n), and b(n,i) for i > p(n)).
Also, b(n,i) = (n - Sum_{j=1..(i-1)} b(n,j)) mod (i+1).

Formulas added by Brant Jones, Oct 14 2013

A082447 a(n) = the number k such that s(k)=0 where s(0)=n and s(i)=s(i-1)-(s(i-1) modulo (i+1)).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15

Offset: 1

Benoit Cloitre, Apr 25 2003

nonn

a(n+1) = number of Mancala numbers <= n, see A007952; n occurs A028913(n-1) times consecutively. - Reinhard Zumkeller, Jun 21 2008

a(n) = number of ones <= n in A130747; see also A002491. - Reinhard Zumkeller, Jul 01 2009

			For n=4, s(0)=4, 4 ->4-4 mod 1=4 ->4-4 mod 2=4 ->4-4 mod 3=3 ->3-3 mod 4=0, hence s(4)=0 and a(4)=4.
For n=6, s(0)=6, s(1)=6-6 mod 2=6, s(2)=6-6 mod 3=6, s(3)=6-6 mod 4=6-2=4, s(4)=4-4 mod 5=0, hence a(6)=4.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A073047, A140060, A140061.

Flatten@Table[First@Position[Rest@FoldList[#1-Mod[#1,#2]&,i,Range[2,i+1]],0], {i,30}] (* Birkas Gyorgy, Feb 26 2011 *)

a(n)=if(n<1, 0, s=n; c=1; while(s-s%c>0, s=s-s%c; c++); c--) \\ corrected by Dan Dima, Jan 18 2025

Conjecture: a(n) = sqrt(Pi*n) + O(1)

a(n) = A073047(n) - 1.

Name corrected by Dan Dima, Jan 18 2025

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

Original entry on oeis.org

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

N. J. A. Sloane

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

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

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.

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

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 *)

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

More terms from Erich Friedman

A028932 Triangular array of winning positions in Tchoukaillon (or Mancala) solitaire.

Original entry on oeis.org

0, 1, 2, 0, 2, 1, 3, 1, 0, 3, 1, 1, 4, 2, 0, 0, 4, 2, 0, 1, 4, 2, 2, 0, 4, 2, 2, 1, 5, 3, 1, 1, 0, 5, 3, 1, 1, 1, 6, 4, 2, 0, 0, 0, 6, 4, 2, 0, 0, 1, 6, 4, 2, 0, 2, 0, 6, 4, 2, 0, 2, 1, 6, 4, 2, 3, 1, 0, 6, 4, 2, 3, 1, 1, 7, 5, 3, 1, 2, 0, 0, 7, 5, 3, 1, 2, 0, 1

Offset: 0

N. J. A. Sloane

The 300 names of Mancala: Abalala'e, Abanga, Abangah, Abouga, Achara, Adi, Adita ta, Adito, Adji, Adjiboto, Adjika, Adji pre, Adjito, Aghi, Agi, Aji, Ajwa, Ale, Andot, Annana, Anywoli, Awale, Awale, Aware, Awari, Awele, Awele Ayo, Ayo Ayo, Azigo,
Ba-Awa, Banga, Bao, Bao kiswahili, Bao solo, Bare, Baruma, Bau, Bawo, Bechi, Boke, Bosh, Bouberoukou, Bouri, Chanka, Chisolo, Chongkak, Choro, Chouba, Chuba, Chunca, Cisolo, Congkak, Coo, Coro, Coro bawo,
Dabuda, dakon, Dakoun, Dara, Darra, Deka, Djonghok, Djonglak, Dwong, erherhe, Endodoi, Enkeshui, Eson xorgol, Esson, eu leu, Fangaya, Fuva, Gabata, Gabatta, Galatjang, Gamacha, Gbegele, Gebta, Gelo, Gepeta, Gesuwa, Gilberta, Giuthi,
Halusa, Hus, Igisoro, Ikiokoto, Imbelece, Imbwe, Impere, Isafu, Ise onzin egbe, Isofu, Isolo, J'erin, jodu, J'odu, Jukuru, Kachig, Ka ia, Kalah, Kalaha, Kalak, Kale, Kalimanta, Kasonko, Katra, Kboo, Kenji guki, Kiarabu, Kisolo, Kiswahilibao, Kiuthi, Kpo, !Krour, Kubuguza,
La'b hakim, La'b madjunni, La'b roseya, Lahemay walida, Lami, Lamlameta, Lamosh, Lam waladach, Langa holo, Layo, leka, Lela, Leyla gobale, Lien, Lizolo, L'ob akila, Lonbeu a cha, Lontu Holo, Luzolo,
Mancala, Mandiare, Manga, Mangala, Mangola, Mankala, Manqala, Manquala, Marabout, Marany, Maruba, Mazageb, Mbangi, Mbau, Mbelete, Mbere, Mbo, Mbothe, Mefuhva, Meusueb, Mewelad, Mofuba, Moro gbegele, Motiq, Msuwa, Mulabalaba, Mungala, Mutiteba, Mwambalula, Mweiso, Mweso,
Nakabile, Nambayi, Naranj, Ncholokoto, Nchomvwa, Nchuba, Nchuwa, Ndoto, Ngar, Njombwa, Nocholokoto, Nsolo, Nsumbi, Ntchuwa, Nummun,
Oko, Olinda, Okwe, Omweeso, Omweso, Otep, Otjitoko, Ot jun, Otra, Ot tjin, Otu, Oure, Ouri, Ourin, Ourre, Ourri, Oware, Owela, Palankuli, Pallamkurie, Pallam kuzhi, Pallanguli, Pallankuli, Pandi, Papadakon, Papandata, Pensur, Pereauni, Peresouni, Poo, Qaluta, Qasuta, Qelat, Ryakati,
Saddeka, Sadeka, Sadiqa, Schach, Serata, Sig, Solo, Sombi, Songo, Soro, Spreta, Sulus ni!shtaw, Sunca, Sungka, Tagega, Tamtam apachi, Tap, Tapata, Tchanka, Tchokajon, Tchonkkak, Tchoukaitlon, Tchukaruma, Tegre, Tjonglak, Toguz xorgol, Toi, Tonka, Topuz xorgol, Tchuba, Tchela, Tshuba, Tshi solo, Tsoro,
Ubao, Ugwasi, Um el bagara, Um el banat, Um el tuweisat, Urdy, Ure, Vai lung thlan, Wale, Walle, Walu, Walya, Ware, Wari, Warri, Wawee, Wawi, Weg, Woaley, Wori, Woribo, Woro, Wouri, Wuli, Wuri, Xorgol, Yada, Yit nuri, Yovodji.

			Triangle begins:
0,
1,
2, 0,
2, 1,
3, 1, 0,
3, 1, 1,
4, 2, 0, 0,
4, 2, 0, 1,
4, 2, 2, 0,
4, 2, 2, 1,
5, 3, 1, 1, 0,
5, 3, 1, 1, 1,
6, 4, 2, 0, 0, 0,
6, 4, 2, 0, 0, 1,
6, 4, 2, 0, 2, 0,
6, 4, 2, 0, 2, 1,
6, 4, 2, 3, 1, 0,
6, 4, 2, 3, 1, 1,
7, 5, 3, 1, 2, 0, 0,
7, 5, 3, 1, 2, 0, 1
...

Y. David, On a sequence generated by a sieving process, Riveon Lematematika, 11 (1957), 26-31.
C. Zaslavsky, Africa Counts: Number and Pattern in Traditional African Culture, Boston: Prindle, Weber and Schmidt, 1973.

D. Betten, Kalahari and the Sequence "Sloane No. 377", Annals Discrete Math., 37, 51-58, 1988.
E. Brisse, African Games Information and Links
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.
D. Eppstein, Combinatorial Game Theory
P. Erdős and E. Jabotinsky, On a sequence of integers generated by a sieving process (Part I), Indagationes Math., 20, 115-123, 1958.
P. Erdős and E. Jabotinsky, On a sequence of integers generated by a sieving process (Part II), Indagationes Math., 20, 124-128, 1958.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
R. W. Wilder, Review of "Africa Counts: Number and Pattern in Traditional African Culture", Historia Mathematica, Vol. 2 (1975), pp. 207-210.
C. Zaslavsky, African and Multicultural Mathematics
Index entries for sequences generated by sieves

Cf. A002491, A007952, A028920, A028931, A028933.

s[list_] := Module[{x = Append[list, 0], i = 1}, While[x[[i]] =!= 0, x[[i]] = x[[i]] - 1; i = i + 1]; x[[i]] = i;If[Last@x == 0, Most[x], x]]; Flatten[Reverse /@ NestList[s, {}, 20]] (* Birkas Gyorgy, Feb 26 2011 *)

To get the next row in the triangle, find the rightmost zero entry in the current row (which may be to the left of the existing entries). In this zero is in position k (counting from the right), change it from 0 to k and subtract 1 from all the entries to its right.

Additional references from Labos Elemer, Nov 07 2000

Revised by N. J. A. Sloane, Jul 16 2012

cp's OEIS Frontend

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A000960 Flavius Josephus's sieve: Start with the natural numbers; at the k-th sieving step, remove every (k+1)-st term of the sequence remaining after the (k-1)-st sieving step; iterate.

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A007952 Generated by a sieve: keep first number, drop every 2nd, keep first, drop every 3rd, keep first, drop every 4th, etc.

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A028920 Pit harvesting sequence for winning solitaire Tchoukaillon (or Mancala).

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A112564 Square array, read by ascending antidiagonals, where each row is a generalized Flavius Josephus sieve (A000960).

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A112565 Main diagonal of square table A112564 of generalized Flavius Josephus sieves.

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A028933 Table of winning positions in Tchoukaillon (or Mancala) solitaire.

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