A364312 -id:A364312 - cp's OEIS frontend

A364313 Length of row n of the irregular triangle A364312.

2, 2, 7, 13, 44, 95, 231

Offset: 1

Wolfdieter Lang, Jul 19 2023

This gives the total number a(n) of nonnegative coefficients of the integer polynomials of degree k = 1, 2, ..., n-1 of Cantor's height n, for n >= 2, and for n = 1 the degree is k = 1, with polynomial 1*x.

Not all rows of A364312 have entries for k = n-1, e.g., for n = 4 the k = 3 entry [1, 0, 0, 1] is not recorded because both x^3 + 1 and the signed version x^3 - 1 factorize. Similar cases appear for n = 6 and n = 7.

			a(3) = 7 because row n = 3 of A364312 is 2, 1, 1, 2, 1, 0, 1, from [2, 1], [1, 2]; [1, 0, 1] for the polynomials 2*x + 1, x + 2, x^2 + 1.

Cf. A364312.

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.

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Jul 19 2023

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.

			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.

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

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.

A364314 Number of polynomials (with nonnegative coefficients) of Cantor's height n and degree k (in the range {1, 2, ..., n-1}), for n >= 2; and for n = 1 the degree is k = 1.

Original entry on oeis.org

1, 1, 3, 5, 14, 26, 57

Offset: 1

Wolfdieter Lang, Jul 19 2023

For details on the recorded integer polynomials and their coefficients see A364312.

			a(3) = 3 because the coefficients in A364312 are [2, 1], [1, 2], for
 degree k = 1, and [1, 0, 1], for degree k = 2, and the three polynomials are 2*x + 1, x + 2, and x^2 + 1.
For the counting of algebraic numbers one also has to use the signed versions with leading sign +, and consider only irreducible polynomials. Therefore, if only real algebraic numbers are considered, [1, 0, 1] does not qualify, because it leads to a pair of complex conjugate roots, and the signed version [1, 0, -1] gives a reducible polynomial.

Cf. A364312, A364313.

A364316 Number of real algebraic numbers of Cantor's height n.

Original entry on oeis.org

1, 2, 4, 12, 28, 72, 172

Offset: 1

Wolfdieter Lang, Jul 19 2023

This gives the row sums of A364315.

See the irregular triangles A364312 for details, and the Cantor references.

Cf. A364312, A364315.

cp's OEIS Frontend

A364313 Length of row n of the irregular triangle A364312.

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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.

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A364314 Number of polynomials (with nonnegative coefficients) of Cantor's height n and degree k (in the range {1, 2, ..., n-1}), for n >= 2; and for n = 1 the degree is k = 1.

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A364316 Number of real algebraic numbers of Cantor's height n.

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