A282615 Number of self-conjugate separable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).

0, 1, 1, 3, 4, 9, 20, 35, 102, 160, 736, 930, 5972, 6766, 59017, 61814, 671651

Offset

1,4

Comments

An inseparable solution is one in which "there is no j such that the first j of the triples are a partition of 1, ..., 3j" (see A202705).

A self-conjugate solution is one in which for every triple (a, b, c) in the partition there exists a "conjugate" triple (m-a, m-b, m-c) or (m-b, m-a, m-c) where m = 3n+1.

| separable | inseparable | either |

-------------------+-----------+-------------+---------+

self-conjugate | A282615 | A279197 | A282616 |

non-self-conjugate | A282618 | A282617 | A282619 |

either | A279199 | A202705 | A104429 |

Links

Table of n, a(n) for n=1..17.

Formula

a(n) = A282616(n) - A279197(n).

a(n) = A279199(n) - A282618(n).

Example

For n = 4 the a(4) = 3 solutions are:
(10,12,11),(7,9,8),(4,6,5),(1,3,2),
(10,12,11),(5,9,7),(4,8,6),(1,3,2), and
(8,12,10),(7,11,9),(2,6,4),(1,5,3).

Crossrefs

Cf. A104429, A202705, A279197, A279199, A282616, A282617, A282618, A282619.

All of A279197, A279198, A202705, A279199, A104429, A282615 are concerned with counting solutions to X+Y=2Z in various ways.

Sequence in context: A291532 A110810 A247579 * A049978 A324764 A092763

Adjacent sequences: A282612 A282613 A282614 * A282616 A282617 A282618

Keyword

nonn,more

Author

Peter Kagey, Feb 19 2017

Extensions

a(11)-a(16) from Fausto A. C. Cariboni, Feb 27 2017

a(17) from Fausto A. C. Cariboni, Mar 22 2017

Status

approved