A364315 Irregular triangle T read by rows obtained from A364312. Row n gives the number of real algebraic numbers from the (also signed) polynomials of Cantor's height n, and degree k, for k = 1, 2, ..., n-1, for n >= 2, and for n = 1 the degree is 1.

1, 2, 4, 0, 4, 8, 0, 8, 8, 12, 0, 4, 32, 20, 16, 0, 12, 28, 100, 16, 16, 0

Offset: 1

Author

Wolfdieter Lang, Jul 19 2023

Keywords

• nonn
• tabf
• more

Comments

The length of row n is A028310(n-1), i.e., 1 for n = 1, and n-1 for n >= 2.

For the nonnegative coefficients of the qualifying polynomials see A364312.

Not all polynomials listed in A364312 lead to real roots. E.g., for n = 3 the entry [1, 0, 1] for k = 2, for polynomial x^2 + 1, has only a pair of complex conjugate roots, and x^2 - 1 is reducible over the integers.

The polynomials listed (by their coefficients) in A364312 which are reducible over the integers have at least one irreducible signed version. E.g., n = 5, k = 2, [1, 2, 1] (with polynomial (x+1)^2), but [1, -2, -1] and [1, 2, -1] do not factor over the integers.

For n >= 3 there are no real roots for k = n-1, if there is an entry in A364312 at all. E.g., for n = 4 there is no entry for k = 3, because x^3 + 1 and x^3 - 1 factorize over the integers. Similar cases appear for n = 6 and 7.

Examples

			The irregular triangle begins:                Row sums A364316(n)
n\k  1  2   3  4  5 6 ...
1:   1                                                   1
2:   2                                                   2
3:   4  0                                                4
4:   4  8   0                                           12
5:   8  8  12  0                                        28
6:   4 32  20 16  0                                     72
7:  12 28 100 16 16 0                                  172
...
T(3, 1) = 4 from [2, 1] and [1, 2], i.e., 2*x + 1, 2*x - 1 and x + 2 and x - 2, giving the 4 real roots -1/2, 1/2, -2, 2.
T(3, 2) = 0, see the third comment above.
T(4, 1) = 4 from [3, 1], [3, -1], [1, 3], [1, -3] giving the 4 real roots -1/3, +1/3, -3, 3.
T(4, 2) = 8 from [2, 0, 1], [1, 0, 2] and [1, 1, 1], with certain signed versions. See the example in A364312.
		

Crossrefs

Cf. A028310, A364312, A364313, A364314, A364316.

Formula

T(n, k) equals the number of real algebraic integers of Cantor's height n and degree k of the irreducible integer polynomials (also signed) obtained from A364312.