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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.

LINKS

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

Matthias Beck and Thomas Zaslavsky, Six Little Squares and How their Numbers Grow, Journal of Integer Sequences, 13 (2010), #10.6.2.

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

.

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).

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 *)

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, R. 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 December 12 20:12 EST 2019. Contains 329961 sequences. (Running on oeis4.)