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A108235 Number of partitions of {1,2,...,3n} into n triples (X,Y,Z) each satisfying X+Y=Z. 9
1, 1, 0, 0, 8, 21, 0, 0, 3040, 20505, 0, 0, 10567748, 103372655, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

a(0)=1 by convention.

REFERENCES

M. Beck and T. Zaslavsky, Six little squares and how their numbers grow, J. Integer Sequences, 13 (2010), #10.6.2.

LINKS

Table of n, a(n) for n=0..15.

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

Wikipedia, Dancing Links

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.

EXAMPLE

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

PROG

(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

CROSSREFS

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

Sequence in context: A221067 A217018 A079386 * A130021 A003864 A182602

Adjacent sequences:  A108232 A108233 A108234 * A108236 A108237 A108238

KEYWORD

nonn,more

AUTHOR

N. J. A. Sloane, Feb 10 2010, based on posting to the Sequence Fans Mailing List by Franklin T. Adams-Watters, Richard K. Guy, R. H. Hardin, Alois P. Heinz, Andrew Weimholt and others.

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

STATUS

approved

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Last modified May 28 16:24 EDT 2017. Contains 287241 sequences.