A338429 -id:A338429 - cp's OEIS frontend

A350050 a(n) = (2n^4 - 6(-1)^nn^2 - 2n^2 + 3*(-1)^n - 3)/96.

0, 0, 0, 2, 4, 14, 24, 52, 80, 140, 200, 310, 420, 602, 784, 1064, 1344, 1752, 2160, 2730, 3300, 4070, 4840, 5852, 6864, 8164, 9464, 11102, 12740, 14770, 16800, 19280, 21760, 24752, 27744, 31314, 34884, 39102, 43320, 48260, 53200, 58940, 64680, 71302, 77924, 85514

Offset: 0

Stefano Spezia, Dec 11 2021

Definitions: (Start)
The k-th exterior power of a vector space V of dimension n is a vector subspace spanned by elements, called k-vectors, that are the exterior product of k vectors v_i in V.
Given a square matrix A that describes the vectors v_i in terms of a basis of V, the k-th exterior power of the matrix A is the matrix that represents the k-vectors in terms of the basis of V. (End)
Conjectures: (Start)
For n > 1, a(n) is the absolute value of the trace of the 2nd exterior power of an n X n square matrix A(n) defined as A[i,j,n] = n - abs((n + 1)/2 - j) - abs((n + 1)/2 - i) (see A349107). Equivalently, a(n) is the absolute value of the coefficient of the term [x^(n-2)] in the characteristic polynomial of the matrix A(n), or the absolute value of the sum of all principal minors of A(n) of size 2.
For k > 2, the trace of the k-th exterior power of the matrix A(n) is equal to zero. (End)
The same conjectures hold for an n X n square matrix A(n) defined as A[i,j,n] = (n mod 2) + abs((n + 1)/2 - j) + abs((n + 1)/2 - i) (see A349108).

Winston de Greef, Table of n, a(n) for n = 0..10000
Wikipedia, Characteristic polynomial
Wikipedia, Exterior algebra
Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Cf. A000982 (trace of matrix A(n)), A317614 (elements sum of matrix A(n)), A349107, A349108.

Cf. A006325, A112742, A338429.

Table[(2*n^4-6*(-1)^n*n^2-2*n^2+3*(-1)^n-3)/96,{n,0,45}]

a(n) = (2*n^4 - 6*(-1)^n*n^2 - 2*n^2 + 3*(-1)^n - 3)/96 \\ Winston de Greef, Jan 28 2024

O.g.f.: 2*x^3*(1 + x^2)/((1 - x)^5*(1 + x)^3).
E.g.f.: (x*(x^3 + 6*x^2 + 3*x + 3)*cosh(x) + (x^4 + 6*x^3 + 9*x^2 - 3*x - 3)*sinh(x))/48.
a(n) = 2*a(n-1) + 2*a(n-2) - 6*a(n-3) + 6*a(n-5) - 2*a(n-6) - 2*a(n-7) + a(n-8) for n > 7.
a(n) = A338429(n-2)/2 for n > 2.
a(2*n-1) = 2*A006325(n).
a(2*n) = A112742(n).
Sum_{n>2} 1/a(n) = (45 - 2*Pi^2 - 4*sqrt(3)*Pi*tanh(sqrt(3)*Pi/2))/4 = 0.920755957767250147865...

A337939 Irregular triangle T(n, m) read by rows: row n gives the distinct length ratios diagonal/side of regular n-gons, DSR(n, k), for n >= 2, k = 1, 2, ..., floor(n/2), expressed by the coefficients in the power basis of the Galois group Gal(Q(rho(n))/Q), where rho(n) = 2*cos(Pi/n), for n >= 2. T(1, 1) is set to 1.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, -1, 0, 1, 1, 0, 1, -1, 0, 1, 0, -2, 0, 1, 1, 0, 1, -1, 0, 1, 1, 1, 1, 0, 1, -1, 0, 1, 0, -2, 0, 1, -4, 0, 2, 1, 0, 1, -1, 0, 1, 0, -2, 0, 1, 1, 0, -3, 0, 1, 1, 0, 1, -1, 0, 1, 0, -2, 0, 1, 0, 0, 1, 0, 2

Offset: 1

Wolfdieter Lang, Jan 15 2021

The length of row n is given in A338431(n), for n >= 1.
The length of the sublists t(n, k) of the power basis coefficients of DSR(n, k), for k = 1, 2, ..., floor(n/2), is 1 if n = 1, for n >= 2 it is k except for n = n(j) = A111774(j) for which the final A219839(n) sublists have fewer than k members.
Trailing vanishing coefficients of the delta(n) = A055034(n) power base elements <1 = rho(n)^0, rho(n)^1, ..., rho(n)^{delta(n)-1}> are not recorded. The coefficients of the minimal polynomial C(n, x) of rho(n) = 2*cos(Pi/n) of degree delta(n) are given in A187360. C(n, rho(n)) = 0 is used to eliminate all powers of rho with exponent >= delta(n).
The length ratios DSR(n, k) := diagonal(n, k)/side(n) of regular n-gons, for n >= 2, and k = 1, 2, ..., floor(n/2) (distinct diagonals, starting with the side for k = 1, in increasing order) are given by DSR(n, k) = S(k-1, rho(n)), with the Chebyshev S polynomials (A049310). See the W. Lang link.
For n = 2, the degenerate case, diagonal/side = side/side = 1 for k = 1. For n = 1 (a point) diagonal/side is undetermined, and T(1, 1) is set to 1.
For the power basis sublists t(n, k), for k = 1, 2, ..., delta(n), only the k coefficients of S(k-1, x) are present (trailing vanishing coefficients are not recorded). For k = delta(n)+1, ..., floor(n/2) less than k coefficients appear due to elimination via C(n, rho(n)) = 0. E.g., for n = 6 with delta(6) = 2 the only coefficient for k = 3 is 2 (coefficient of rho^0). This appears for n = n(j) = A111774(j), because then floor(n/2) - delta(n) = A219839(n) > 0.
Because A219839(n) = 0 means that n is from A174090, i.e., a prime or a power of 2 (complement of A111774), these rows n have all the sublists t(n, k) with the k coefficients of S(k-1, x), hence they are identical (but the basis differs). See especially the table for the pairs of consecutive numbers n with identical coefficients, like (2, 3), (4, 5), (16, 17), (256, 267), (65536, 65537), ?... (cf. Fermat primes A019434).

			The irregular triangle T(n, m) begins: (For n >= 4 the bar divides the DSR(n, k) power basis coefficients, the sublists t(n, k), for k = 1, 2, ..., floor(n/2))
n \ m  1   2 3    4  5  6   7  8 9 10   11 12 12  13 14  15 16 17 18 19 20 ...
1:     1
2:     1
3:     1
4:     1 | 0 1
5:     1 | 0 1
6:     1 | 0 1 |  2
7:     1 | 0 1 | -1  0  1
8:     1 | 0 1 | -1  0  1 | 0 -2 0  1
9:     1 | 0 1 | -1  0  1 | 1  1
10:    1 | 0 1 | -1  0  1 | 0 -2 0  1 | -4  0  2
11:    1 | 0 1 | -1  0  1 | 0 -2 0  1 |  1  0 -3   0  1
12:    1 | 0 1 | -1  0  1 | 0 -2 0  1 |  0  0  1 | 0  2
13:    1 | 0 1 | -1  0  1 | 0 -2 0  1 |  1  0 -3   0  1 | 0  3  0 -4  0  1
...
n = 14: 1 | 0 1 | -1 0 1 | 0 -2 0 1 | 1 0 -3 0 1 | 0 3 0 -4 0 1 | 6 0 -8 0 2,
n = 15: 1 | 0 1 | -1 0 1 | 0 -2 0 1 | 0 4 1 -1 | 1 -2 0 1 | -1 1 1,
n = 16 and n = 17: 1 | 0 1 | -1  0 1 | 0 -2 0 1 | 1 0 -3 0 1 | 0 3 0 -4 0 1 | -1 0 6 0 -5 0 1 | 0 -4 0 10 0 -6 0 1,
--------------------------------------------------------------------------------
n = 5: DSR(5, 1) = 1 = side(5)/side(5), DSR(5, 2) = 1*rho(5) = A001622 (golden section).
n = 8: DSR(8, 1) = 1 = side(8)/side(8), DSR(8, 2) = 1*rho(8) = sqrt(2+sqrt(2)) = A179260, DSR(8, 3) = -1 + rho(8)^2 = 1 + sqrt(2) = A014176, DSR(8, 4) = -2*rho(8) + 1*rho(8)^3 = sqrt(2)*rho(8) = A121601.

Wolfdieter Lang, The field Q(2cos(pi/n)), its Galois group and length ratios in the regular n-gon, arXiv:1210.1018 [math.GR], 2012-2017.

Cf. A001622, A055034, A014176, A049310, A019434, A111774, A121601, A179260, A187360, A219839, A338429, A337940, A338431.

T(1, 1) = 1, and in row n, for n >= 2, the power base coefficients of Gal(Q(2*cos(Pi/n))/Q) for DSR(n, k) := diagonal(n, k)/side(n) of regular n-gons, for k = 1, 2, ..., floor(n/2), are listed as t(n, k) in this order, with trailing vanishing coefficients omitted.

cp's OEIS Frontend

A350050 a(n) = (2n^4 - 6(-1)^nn^2 - 2n^2 + 3*(-1)^n - 3)/96.

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cp's OEIS Frontend

A350050 a(n) = (2*n^4 - 6*(-1)^n*n^2 - 2*n^2 + 3*(-1)^n - 3)/96.

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A350050 a(n) = (2n^4 - 6(-1)^nn^2 - 2n^2 + 3*(-1)^n - 3)/96.