cp's OEIS Frontend

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

%I A282615 #35 Jul 15 2025 08:59:19
%S A282615 0,1,1,3,4,9,20,35,102,160,736,930,5972,6766,59017,61814,671651,
%T A282615 675379,8844028,8675583,130880467,126385830,2163551657,2049560059,
%U A282615 39112954305,36883483406,768337929193,720918897940,16279025598443,15303083773040,373743187469167,349148771223261,9095126347788632
%N A282615 Number of self-conjugate separable solutions of X + Y = 2Z (integer, disjoint triples from {1,2,3,...,3n}).
%C A282615 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).
%C A282615 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.
%C A282615                    | separable | inseparable | either  |
%C A282615 -------------------+-----------+-------------+---------+
%C A282615 self-conjugate     | A282615   | A279197     | A282616 |
%C A282615 non-self-conjugate | A282618   | A282617     | A282619 |
%C A282615 either             | A279199   | A202705     | A104429 |
%F A282615 a(n) = A282616(n) - A279197(n).
%F A282615 a(n) = A279199(n) - A282618(n).
%F A282615 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
%e A282615 For n = 4 the a(4) = 3 solutions are:
%e A282615   (10,12,11),(7,9,8),(4,6,5),(1,3,2),
%e A282615   (10,12,11),(5,9,7),(4,8,6),(1,3,2), and
%e A282615   (8,12,10),(7,11,9),(2,6,4),(1,5,3).
%Y A282615 Cf. A104429, A202705, A279197, A279199, A282616, A282617, A282618, A282619.
%Y A282615 All of A279197, A279198, A202705, A279199, A104429, A282615 are concerned with counting solutions to X+Y=2Z in various ways.
%K A282615 nonn
%O A282615 1,4
%A A282615 _Peter Kagey_, Feb 19 2017
%E A282615 a(11)-a(16) from _Fausto A. C. Cariboni_, Feb 27 2017
%E A282615 a(17) from _Fausto A. C. Cariboni_, Mar 22 2017
%E A282615 a(18)-a(24) from _Bert Dobbelaere_, May 29 2025
%E A282615 a(25)-a(33) from _Martin Fuller_, Jul 15 2025