cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

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Author

N. J. A. Sloane

Keywords

Examples

			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

References

  • 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).

Links

Crossrefs

Cf. A002848, A108235, A161826.

Programs

  • PARI
    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

Extensions

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

Views

Author

N. J. A. Sloane

Keywords

Examples

			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.

References

  • 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).

Links

Crossrefs

Cf. A002849, A108235, A161826.

Formula

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

Extensions

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

A108235 Number of partitions of {1,2,...,3n} into n triples (X,Y,Z) each satisfying X+Y=Z.

Original entry on oeis.org

1, 1, 0, 0, 8, 21, 0, 0, 3040, 20505, 0, 0, 10567748, 103372655, 0, 0, 142664107305, 1836652173363, 0, 0
Offset: 0

Views

Author

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 and others

Keywords

Comments

a(0)=1 by convention.

Examples

			For m = 1 the unique solution is 1 + 2 = 3.
For m = 4 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
.
The 8 solutions for m = 4, one per line:
  (1,  5,  6), (2,  8, 10), (3,  9, 12), (4,  7, 11);
  (1,  5,  6), (2,  9, 11), (3,  7, 10), (4,  8, 12);
  (1, 10, 11), (2,  5,  7), (3,  6,  9), (4,  8, 12);
  (1,  6,  7), (2, 10, 12), (3,  8, 11), (4,  5,  9);
  (1,  9, 10), (2,  4,  6), (3,  8, 11), (5,  7, 12);
  (1, 11, 12), (2,  6,  8), (3,  7, 10), (4,  5,  9);
  (1,  8,  9), (2, 10, 12), (3,  4,  7), (5,  6, 11);
  (1, 11, 12), (2,  7,  9), (3,  5,  8), (4,  6, 10).

Links

Crossrefs

Cf. A002848, A002849, A161826, A202951, A202952.

Programs

  • Mathematica
    Table[Length[Select[Subsets[Select[Subsets[Range[3 n], {3}], #[[1]] + #[[2]] == #[[3]] &], {n}], Range[3 n] == Sort[Flatten[#]] &]], {n, 0,
5}]  (* Suitable only for n<6. See Knuth's Dancing Links algorithm for n>5. *) (* Robert Price, Apr 03 2019 *)
  • Sage
    A = lambda n:sum(1 for t in DLXCPP([(a-1,b-1,a+b-1) for a in (1..3*n) for b in (1..min(3*n-a,a-1))])) # Tomas Boothby, Oct 11 2013

Formula

a(n) = 0 unless n == 0 or 1 (mod 4). For n == 0 or 1 (mod 4), a(n) = A002849(3n). See A002849 for references and further information.

Extensions

a(12) from R. H. Hardin, Feb 11 2010
a(12) confirmed and a(13) computed (using Knuth's dancing links algorithm) by Alois P. Heinz, Feb 11 2010
a(13) confirmed by Tomas Boothby, Oct 11 2013
a(16) from Frank Niedermeyer, Apr 19 2020
a(17)-a(19) from Frank Niedermeyer, May 02 2020
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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