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A358655 a(n) is the number of distinct scalar products which can be formed by pairs of signed permutations (V, W) of [n].

%I A358655 #54 Oct 02 2024 07:29:01
%S A358655 1,2,7,24,61,111,183,281,409,571,771,1013,1301,1639,2031,2481,2993,
%T A358655 3571,4219,4941,5741,6623,7591,8649,9801,11051,12403,13861,15429,
%U A358655 17111,18911,20833,22881,25059,27371,29821,32413,35151,38039,41081,44281,47643
%N A358655 a(n) is the number of distinct scalar products which can be formed by pairs of signed permutations (V, W) of [n].
%C A358655 Let V be an n-vector of the numbers 1 to n in order and let W be an n-vector of any signed permutation of these numbers. Numbers in W may be either positive or negative. a(n) is the number of different values for the scalar product V*W for all possible W. We allow all combinations of positive and negative signs in W.
%C A358655 Another interpretation of this sequence: A signed permutohedron is also called the Coxeter permutohedron of the family C_n and has A000165(n) vertices. If we choose a vector of one vertex of such a permutohedron to the origin, and cut this permutohedron in slices by hyperplanes which are orthogonal to this vector such that in each slice lies at least one vertex of this permutohedron, then a(n) is the count of such slices obtained by this process.
%C A358655 a(n) is odd if the plane through the origin is occupied by vertices, this means A358629(n) > 0.
%C A358655 For n > 3 all possible planes are occupied by vertices and thus a nice formula ( see formula section ) exists.
%C A358655 The permutohedra for small n are:
%C A358655   n = 2 Octagon.
%C A358655   n = 3 Truncated cuboctahedron.
%C A358655   n = 4 Omnitruncated tesseract.
%C A358655   n = 5 Omnitruncated 5-cube.
%H A358655 Paolo Xausa, <a href="/A358655/b358655.txt">Table of n, a(n) for n = 0..10000</a>
%H A358655 Thomas Scheuerle, <a href="/A358655/a358655.png">The permutohedron for a(3) a truncated cuboctahedron</a>.
%H A358655 Thomas Scheuerle, <a href="/A358655/a358655_1.png">The truncated cuboctahedron of a(3) with 24 planes intersecting in at least one vertex</a>.
%H A358655 Thomas Scheuerle, <a href="/A358655/a358655_2.png">The octagon of a(2) with 7 planes intersecting in at least one vertex</a>.
%H A358655 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A358655 a(n) = (2*n^3 + 3*n^2 + n + 3)/3 = A188475(n), for n > 3 (because valid if A000165(n)/2 > A188475(n)).
%F A358655 From _Stefano Spezia_, Nov 28 2022: (Start)
%F A358655 G.f.: (1 - 2*x + 5*x^2 + 4*x^3 - 15*x^5 + 16*x^6 - 5*x^7)/(1 - x)^4.
%F A358655 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4) for n > 7. (End)
%e A358655 a(2) = 7
%e A358655 Columns in the table below:
%e A358655 A: Result of the scalar product.
%e A358655 B: Count of combinations for this result.
%e A358655 C: An example.
%e A358655     A  B      C
%e A358655     5  1  [ 2,  1]*[ 2, 1]
%e A358655     4  1  [ 1,  2]*[ 2, 1]
%e A358655     3  1  [ 2, -1]*[ 2, 1]
%e A358655     0  2  [ 1, -2]*[ 2, 1]
%e A358655    -3  1  [-2,  1]*[ 2, 1]
%e A358655    -4  1  [-1, -2]*[ 2, 1]
%e A358655    -5  1  [-2, -1]*[ 2, 1]
%e A358655 We have 7 rows. The sum over B is A000165(2).
%e A358655 For a(2) all vectors C are part of the vertices of an octagon.
%t A358655 LinearRecurrence[{4, -6, 4, -1}, {1, 2, 7, 24, 61, 111, 183, 281}, 50] (* _Paolo Xausa_, Oct 02 2024 *)
%Y A358655 Cf. A000165, A188475, A358629.
%K A358655 nonn,easy
%O A358655 0,2
%A A358655 _Thomas Scheuerle_, Nov 25 2022