A002848 - OEIS

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.
(Formerly M0295 N0106)

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,6

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

Table of n, a(n) for n=0..44.

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.

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

EXAMPLE

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.

CROSSREFS

Cf. A002849, A108235, A161826.

Sequence in context: A060357 A064714 A152255 * A032257 A038075 A032236

Adjacent sequences: A002845 A002846 A002847 * A002849 A002850 A002851

KEYWORD

nonn

AUTHOR

N. J. A. Sloane

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

STATUS

approved