cp's OEIS Frontend

A294172 Maximum value of the cyclic convolution of the first n positive integers with themselves.

%I A294172 #33 Oct 14 2024 11:22:04
%S A294172 1,5,13,28,50,83,126,184,255,345,451,580,728,903,1100,1328,1581,1869,
%T A294172 2185,2540,2926,3355,3818,4328,4875,5473,6111,6804,7540,8335,9176,
%U A294172 10080,11033,12053,13125,14268,15466,16739,18070,19480,20951,22505,24123,25828,27600
%N A294172 Maximum value of the cyclic convolution of the first n positive integers with themselves.
%C A294172 Conjecture: a(n) = (n*(13 + 3*(-1)^n + 24*n + 14*n^2))/48, and then lim_{n -> infinity} a(n)/n^3 = 7/24.
%C A294172 The conjectured formula is true. See links. - _Sela Fried_, Aug 13 2024.
%H A294172 Sela Fried, <a href="/A294172/a294172.pdf">On the maximum value of the cyclic convolution of the first n positive integers with themselves (A294172)</a>, 2024.
%H A294172 Sela Fried, <a href="https://arxiv.org/abs/2410.07237">Proofs of some Conjectures from the OEIS</a>, arXiv:2410.07237 [math.NT], 2024. See p. 8.
%F A294172 a(n) = Max {x; x=Sum_{i=1..n}(n-i+1)*(1+(i+k) mod n); for k=1..n}.
%F A294172 Conjectures from _Colin Barker_, Feb 11 2018: (Start)
%F A294172 G.f.: x*(1 + 3*x + 2*x^2 + x^3) / ((1 - x)^4*(1 + x)^2).
%F A294172 a(n) = 2*a(n-1) + a(n-2) - 4*a(n-3) + a(n-4) + 2*a(n-5) - a(n-6) for n>6.
%F A294172 (End)
%e A294172 For n = 4, the four possible cyclic convolutions of the first four positive integers with themselves are:
%e A294172 (1,2,3,4).(4,3,2,1) = 1*4 + 2*3 + 3*2 + 4*1 = 4 + 6 + 6 + 4 = 20,
%e A294172 (1,2,3,4).(3,2,1,4) = 1*3 + 2*2 + 3*1 + 4*4 = 3 + 4 + 3 + 16 = 26,
%e A294172 (1,2,3,4).(2,1,4,3) = 1*2 + 2*1 + 3*4 + 4*3 = 2 + 2 + 12 + 12 = 28,
%e A294172 (1,2,3,4).(1,4,3,2) = 1*1 + 2*4 + 3*3 + 4*2 = 1 + 8 + 9 + 8 = 26,
%e A294172 then a(4)=28 because 28 is the maximum among the four values.
%t A294172 a[n_] := Max[Table[Range[n].RotateRight[Reverse[Range[n]], k], {k, 0, n - 1}]];
%t A294172 Table[a[n], {n, 1, 45}]
%o A294172 (PARI) a(n) = vecmax(vector(n, k, sum(i=1, n, (n-i+1)*(1+(i+k) % n)))); \\ _Michel Marcus_, Feb 11 2018
%Y A294172 Cf. A000292, A299111.
%K A294172 nonn
%O A294172 1,2
%A A294172 _Andres Cicuttin_, Feb 10 2018