A108235 -id:A108235 - cp's OEIS frontend

A202952 A108235(n)-A202951(n).

Original entry on oeis.org

0, 0, 0, 0, 2, 11, 0, 0, 2300

Offset: 0

N. J. A. Sloane, Dec 26 2011

R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]

Cf. A108235, A202951.

A284737 A condensed version of A108235.

Original entry on oeis.org

1, 1, 8, 21, 3040, 20505, 10567748, 103372655

Offset: 1

N. J. A. Sloane, Apr 09 2017

Lists only the terms A108235(n) for n == 0 or 3 mod 12.

See A108235, the main entry for this sequence, for more information.

R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]
Ana Rechtman, Août 2018, 3e défi, Images des Mathématiques, CNRS, 2018 (in French).

Cf. A108235.

A002849 Number of maximal collections of pairwise disjoint subsets {X,Y,Z} of {1, 2, ..., n}, each satisfying X + Y = Z.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 3, 10, 25, 12, 42, 8, 40, 204, 21, 135, 1002, 4228, 720, 5134, 29546, 4079, 35533, 3040, 28777, 281504, 20505, 212283, 2352469, 16907265, 1669221, 19424213, 167977344, 14708525, 191825926, 10567748, 149151774, 2102286756, 103372655, 1534969405

Offset: 1

N. J. A. Sloane

nonn

			For n = 3, the unique solution is 1 + 2 = 3.
For n = 12, there are 8 solutions:
  1  5  6 | 1  5  6 | 2  5  7 | 1  6  7
  2  8 10 | 3  7 10 | 3  6  9 | 4  5  9
  4  7 11 | 2  9 11 | 1 10 11 | 3  8 11
  3  9 12 | 4  8 12 | 4  8 12 | 2 10 12
  --------+---------+---------+--------
  2  4  6 | 2  6  8 | 3  4  7 | 3  5  8
  1  9 10 | 4  5  9 | 1  8  9 | 2  7  9
  3  8 11 | 3  7 10 | 5  6 11 | 4  6 10
  5  7 12 | 1 11 12 | 2 10 12 | 1 11 12

R. K. Guy, "Sedlacek's Conjecture on Disjoint Solutions of x+y= z," in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
R. K. Guy, "Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics," in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.
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).

Martin Fuller, Table of n, a(n) for n = 1..52 (terms 1..42 from Fausto A. C. Cariboni, terms 43..44 from Frank Niedermeyer)
R. K. Guy, Letter to N. J. A. Sloane, Jun 24 1971: front, back [Annotated scanned copy, with permission]
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971. [Annotated scanned copy, with permission]
Richard K. Guy, The unity of combinatorics, in Proc. 25th Iran. Math. Conf., Tehran, (1994), Math. Appl. 329 (1994) 129-159, Kluwer Acad. Publ., Dordrecht, 1995.
Christian Hercher and Frank Niedermeyer, Efficient Calculation the Number of Partitions of the Set {1, 2, ..., 3n} into Subsets {x, y, z} Satisfying x + y = z, arXiv:2307.00303 [math.CO], 2023.
Nigel Martin, Solving a conjecture of Sedlacek: maximal edge sets in the 3-uniform sumset hypergraphs, Discrete Mathematics, Volume 125, 1994, pp. 273-277.
Matheplanet Calculating sequence element a(16) of OEIS A108235
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]

Cf. A002848, A108235, A161826.

nxyz(v,t)=local(n,r,x2); r=0; if(t==0,return(1)); for(i3=3*t,#v, n=v[i3]; for(i1=1,i3-2, x2=n-v[i1]; if(x2<=v[i1],break); for(i2=i1+1,i3-1, if(v[i2]>=x2, if(v[i2]==x2, r+=nxyz(vector(i3-3,k,v[if(kFranklin T. Adams-Watters

Edited by N. J. A. Sloane, Feb 10 2010, based on posting to the Sequence Fans Mailing List by Franklin T. Adams-Watters, R. K. Guy, R. H. Hardin, Alois P. Heinz, Andrew Weimholt, Max Alekseyev and others
a(32)-a(39) from Max Alekseyev, Feb 23 2012
Definition corrected by Max Alekseyev, Nov 16 2012, Jul 06 2023
a(40)-a(41) from Fausto A. C. Cariboni, Feb 04 2017
a(42) from Fausto A. C. Cariboni, Mar 12 2017

A002848 Number of maximal collections of pairwise disjoint subsets {X,Y,Z} of {1, 2, ..., n} with X + Y = Z (as in A002849), with the property that n is in one of the subsets.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 7, 15, 12, 30, 8, 32, 164, 21, 114, 867, 3226, 720, 4414, 24412, 4079, 31454, 3040, 25737, 252727, 20505, 191778, 2140186, 14554796, 1669221, 17754992, 148553131, 14708525, 177117401, 10567748, 138584026, 1953134982, 103372655, 1431596750, 22374792451, 218018425976, 16852166906, 254094892254

Offset: 0

N. J. A. Sloane

nonn

			Examples from _Alois P. Heinz_, Feb 12 2010:
A002848(7) = 3:
  [1, 3, 4], [2, 5, 7]
  [1, 5, 6], [3, 4, 7]
  [2, 3, 5], [1, 6, 7]
A002848(8) = 7:
  [1, 3, 4], [2, 6, 8]
  [1, 4, 5], [2, 6, 8]
  [1, 6, 7], [3, 5, 8]
  [2, 3, 5], [1, 7, 8]
  [2, 4, 6], [1, 7, 8]
  [2, 4, 6], [3, 5, 8]
  [3, 4, 7], [2, 6, 8]
A002848(10) = 12:
  [1, 4, 5], [2, 6, 8], [3, 7, 10]
  [1, 4, 5], [3, 6, 9], [2, 8, 10]
  [1, 5, 6], [3, 4, 7], [2, 8, 10]
  [1, 6, 7], [4, 5, 9], [2, 8, 10]
  [1, 7, 8], [2, 3, 5], [4, 6, 10]
  [1, 8, 9], [2, 3, 5], [4, 6, 10]
  [1, 8, 9], [2, 4, 6], [3, 7, 10]
  [1, 8, 9], [2, 5, 7], [4, 6, 10]
  [2, 4, 6], [3, 5, 8], [1, 9, 10]
  [2, 6, 8], [3, 4, 7], [1, 9, 10]
  [2, 6, 8], [4, 5, 9], [3, 7, 10]
  [2, 7, 9], [3, 5, 8], [4, 6, 10]
See A002849 for further examples.

R. K. Guy, "Sedlacek's Conjecture on Disjoint Solutions of x+y= z," in Proc. Conf. Number Theory. Pullman, WA, 1971, pp. 221-223.
R. K. Guy, "Packing [1,n] with solutions of ax + by = cz; the unity of combinatorics," in Colloq. Internaz. Teorie Combinatorie. Rome, 1973, Atti Conv. Lincei. Vol. 17, Part II, pp. 173-179, 1976.
Richard K. Guy, The unity of combinatorics, in Proc. 25th Iran. Math. Conf., Tehran, (1994), Math. Appl. 329 (1994) 129-159, Kluwer Acad. Publ., Dordrecht, 1995.
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).

Martin Fuller, Table of n, a(n) for n = 0..52
R. K. Guy, Letter to N. J. A. Sloane, June 24 1971: front, back [Annotated scanned copy, with permission]
R. K. Guy, Sedlacek's Conjecture on Disjoint Solutions of x+y= z, Univ. Calgary, Dept. Mathematics, Research Paper No. 129, 1971. [Annotated scanned copy, with permission]
Christian Hercher and Frank Niedermeyer, Efficient Calculation the Number of Partitions of the Set {1, 2, ..., 3n} into Subsets {x, y, z} Satisfying x + y = z, arXiv:2307.00303 [math.CO], 2023.
Nigel Martin, Solving a conjecture of Sedlacek: maximal edge sets in the 3-uniform sumset hypergraphs, Discrete Mathematics, Volume 125, 1994, pp. 273-277.

Cf. A002849, A108235, A161826.

For n >= 2, a(n) = A002849(n) if n == 0,3,7,10 (mod 12), otherwise a(n) = A002849(n) - A002849(n-1). - _Franklin T. Adams-Watters; corrected by Max Alekseyev, Jul 06 2023

Edited by N. J. A. Sloane, Feb 10 2010, based on posting to the Sequence Fans Mailing List by Franklin T. Adams-Watters, R. K. Guy, R. H. Hardin, Alois P. Heinz, Andrew Weimholt, Max Alekseyev and others
a(32)-a(39) from Max Alekseyev, Feb 23 2012
Definition corrected by Max Alekseyev, Nov 16 2012, Jul 06 2023
a(40)-a(42) from Fausto A. C. Cariboni, Mar 12 2017
a(43)-a(44) computed from A002849 by Max Alekseyev, Jul 06 2023

A161826 Number of maximal vertex-independent sets in the hypergraph with nodes V = {1, 2, ..., n} and "edges" consisting of the triples (X,Y,Z) with X

Original entry on oeis.org

1, 1, 3, 2, 6, 1, 6, 1, 5, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4

Offset: 1

N. J. A. Sloane, Feb 10 2010

A subset S of V is vertex-independent if there is no edge (X,Y,Z) with X, Y, Z all in S.

Continued fraction expansion of (3452449 + 2*sqrt(2))/1943849. - Stefano Spezia, Mar 17 2024

J. Sedláček, On a set system, Annals New York Acad. Sci., 175 (No. 1, 1970), 329-330.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A002848, A002849, A108235.

a(2k)=1, a(2k+1)=4 for k >= 5.

G.f.: x*(1 + x + 2*x^2 + x^3 + 3*x^4 - x^5 - x^8 - x^10)/((1 - x)*(1 + x)). - Stefano Spezia, Mar 17 2024

A202951 Number of Nickerson-type partitions of [1,...,3n] into triples satisfying x+y=z.

Original entry on oeis.org

1, 1, 0, 0, 6, 10, 0, 0, 700

Offset: 0

N. J. A. Sloane, Dec 26 2011

Perhaps an incorrect version of A004075? Sequence values are from p. 51 of Nowakowski. - Martin Fuller, Jul 06 2025

R. S. Nickerson, Problem E1845, Amer. Math. Monthly, p. 81, 73 (1966).
R. S. Nickerson and D. C. B. Marsh, E1845: A variant of Langford's Problem, American Math. Monthly, 1967, 74, 591-595.
R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.]

Cf. A108235, A202952.

A203435 Number of partitions of {1,2,...,4n} into n 4-element subsets having the same sum.

Original entry on oeis.org

1, 1, 4, 32, 392, 6883, 171088, 5661874, 242038179, 13147317481

Offset: 0

Alois P. Heinz, Jan 01 2012

The element sum of each subset is 8n+2. The larger terms were computed with Knuth's dancing links algorithm.

			a(1) = 1: {1,2,3,4}.
a(2) = 4: {1,2,7,8}, {3,4,5,6}; {1,3,6,8}, {2,4,5,7}; {1,4,5,8}, {2,3,6,7}; {1,4,6,7}, {2,3,5,8}.

Wikipedia, Dancing Links
Wikipedia, Partition of a set

Column k=4 of A203986.

Cf. A104185, A108235, A203017, A264813.

b:= proc() option remember; local i, j, t, m; m:= args[nargs]; if args[1]=0 then `if`(nargs=2, 1, b(args[t] $t=2..nargs)) elif args[1]<1 then 0 else add(`if`(args[j] `if`(n=0, 1, b(((8*n+2)+4/97) $n, 4*n)/n!): seq(a(n), n=0..6);

b[l_] := b[l] = Module[{nl = Length[l], k = l[[-1]], m = l[[-2]]}, Which[l[[1]] == 0, If[nl == 3, 1, b[l[[2 ;; nl]]]], l[[1]] < 1, 0, True, Sum[If[l[[j]] < m, 0, b[Join[Sort[Table[l[[i]] - If[i == j, m + 1/97, 0], {i, 1, nl - 2}]], {m - 1, k}]]], {j, 1, nl - 2}]]];
a[n_] := If[n == 0, 1, b[Join[Array[8*n + 2 + 4/97& , n], {4*n, 4}]]/n!];
Table[a[n], {n, 0, 6}] (* Jean-François Alcover, Jun 03 2018, adapted from Maple *)

A264813 Number of permutations of 3 indistinguishable copies of 1,...,n such that the first and second copies of j are adjacent and there are exactly j numbers between the second and the third copy of j.

Original entry on oeis.org

1, 0, 1, 1, 0, 3, 6, 0, 53, 199, 0, 2908, 13699, 0, 369985, 2135430, 0, 87265700, 611286653, 0

Offset: 0

Alois P. Heinz, Nov 25 2015

a(n) = 0 for n == 1 (mod 3).

			a(0) = 1: the empty permutation.
a(2) = 1: 221121.
a(3) = 1: 223321131.
a(5) = 3: 223325534411514, 225523344531141, 552244253341131.
a(6) = 6: 221121665544336543, 225523366534411614, 225526633544361141, 446611415563322532, 552266253344631141, 665544336543221121.

Eric Weisstein's World of Mathematics, Langford's Problem
Wikipedia, Dancing Links
Wikipedia, Langford pairing

Cf. A014552, A104185, A108235, A176127, A203435, A261516, A261517, A321956.

A202954 Number of partitions of [1,...,3n] into triples satisfying x+y=4z.

Original entry on oeis.org

1, 0, 0, 0, 0, 0, 0, 0, 6, 0, 5, 0, 0, 349, 0, 443, 0, 0, 110757, 0, 1254452, 0, 0, 152965479, 0

Offset: 0

N. J. A. Sloane, Dec 26 2011

a(n)=0 when n is not congruent to 0 or 3 mod 5. - Edward Moody, Jan 17 2021

R. J. Nowakowski, Generalizations of the Langford-Skolem problem, M.S. Thesis, Dept. Math., Univ. Calgary, May 1975. [Scanned copy, with permission.] See Table II.6.

Cf. A108235, A104429.

a(14)-a(17) from Alois P. Heinz, Dec 28 2011
a(18) from Alois P. Heinz, Jan 04 2012
a(19)-a(24) from Edward Moody, Jan 17 2021

cp's OEIS Frontend

A202952 A108235(n)-A202951(n).

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A284737 A condensed version of A108235.

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A002849 Number of maximal collections of pairwise disjoint subsets {X,Y,Z} of {1, 2, ..., n}, each satisfying X + Y = Z.

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A002848 Number of maximal collections of pairwise disjoint subsets {X,Y,Z} of {1, 2, ..., n} with X + Y = Z (as in A002849), with the property that n is in one of the subsets.

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A161826 Number of maximal vertex-independent sets in the hypergraph with nodes V = {1, 2, ..., n} and "edges" consisting of the triples (X,Y,Z) with X

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Formula

A202951 Number of Nickerson-type partitions of [1,...,3n] into triples satisfying x+y=z.

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A203435 Number of partitions of {1,2,...,4n} into n 4-element subsets having the same sum.

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Programs

Maple

Mathematica

A264813 Number of permutations of 3 indistinguishable copies of 1,...,n such that the first and second copies of j are adjacent and there are exactly j numbers between the second and the third copy of j.

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A202954 Number of partitions of [1,...,3n] into triples satisfying x+y=4z.

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Extensions