A282615 - OEIS

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, 675379, 8844028, 8675583, 130880467, 126385830, 2163551657, 2049560059, 39112954305, 36883483406, 768337929193, 720918897940, 16279025598443, 15303083773040, 373743187469167, 349148771223261, 9095126347788632

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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..33.

FORMULA

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

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

a(n) = Sum_{i=1..floor(n/2)} A202705(i) * (A282616(n-2*i) if n>2*i else 1) = Sum_{i=1..floor(n/2)} A104429(i) * (A279197(n-2*i) if n>2*i else 1). - Martin Fuller, Jul 15 2025

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

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

a(18)-a(24) from Bert Dobbelaere, May 29 2025

a(25)-a(33) from Martin Fuller, Jul 15 2025

STATUS

approved