A364314 -id:A364314 - cp's OEIS frontend

A364312 Irregular triangle T read by rows, giving in row n the nonnegative coefficients of polynomials of height n and degree k (of decreasing powers), for k = 1, 2, ..., n-1, used for Cantor's counting of algebraic numbers, written for m = 1, 2, ..., A364313(n), for n >= 2, and for n = 1 the degree is k = 1.

1, 0, 1, 1, 2, 1, 1, 2, 1, 0, 1, 3, 1, 1, 3, 2, 0, 1, 1, 0, 2, 1, 1, 1, 4, 1, 1, 4, 3, 2, 2, 3, 3, 0, 1, 1, 0, 3, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 0, 0, 1, 1, 0, 0, 2, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 1, 5, 1, 1, 5, 4, 0, 1, 1, 0, 4, 3, 0, 2, 2, 0, 3, 3, 1, 1, 1, 3, 1, 1, 1, 3, 2, 2, 1, 2, 1, 2, 1, 2, 2

Offset: 1

Wolfdieter Lang, Jul 19 2023

The length of row n is A364313(n). Different orders k are separated by a semicolon in the examples below.
The number of polynomials given from row n is A364314(n).
The entries for k = n-1 are only present for n >= 2 and A001227(n-1) = 1, that is, n = 2^q + 1 = A000051(q), for q >= 0. This is because otherwise x^(n-1) + 1 and x^(n-1) - 1 are both reducible (factorize over the integers).
For Cantor's counting (and determination) of algebraic numbers these polynomials have later to be signed, keeping the positive leading coefficient. See the example for n = 4 below. Complex solutions are omitted if real algebraic numbers are counted.
The polynomials with nonnegative coefficients recorded here are sometimes reducible over the integers. But in this case irreducible signed versions exist. E.g., for n = 6 and k = 2 the polynomial x^2 + 3*x + 2 = (x + 1)*(x + 2) is recorded as [1,3,2] (falling powers of x), because x^2 + 3*x - 2 and x^2 - 3*x - 2 are irreducible, each having two real solutions.
The number of distinct real solutions of the signed polynomials of degree k and height n is given in A364315(n, k). The total number is A364316(n). Note that no repetition of real solutions already obtained for lower heights can appear due to irreducibility. For the list of all real algebraic numbers for heights 1 to 7 see the W. Lang link.
The coefficients of the polynomials are determined from the relative prime compositions of K = n - (k-1). The order is taken from the corresponding partitions, with rising number of parts m, and for given m the order is anti-lexicographic (e.g., [4,1,1], [3,2,1] for K = 6 and m = 3). For each partition the compositions are ordered also anti-lexicographically, not considering the possible 0 parts which are distributed according to decreasing powers of x (e.g., [3,1,0,1], [3,0,1,1], [1,3,0,1], [1,0,3,1], [1,1,0,3], [1,0,1,3]).

			The irregular triangle T, with entries T(n, m), begins: (increasing k >= 1 values are separated by ;)
n\m   1  2    3  4    5  6  7    8  9 10   11 12 13 ...
1:   [1, 0]
2:   [1, 1]
3:   [2, 1], [1, 2]; [1, 0, 1]
4:   [3, 1], [1, 3]; [2, 0, 1], [1, 0, 2], [1, 1, 1]
...
n = 5: [4, 1], [1, 4], [3, 2], [2, 3]; [3, 0, 1], [1, 0, 3], [2, 1, 1], [1, 2, 1], [1, 1, 2]; [2, 0, 0, 1], [1, 0, 0, 2], [1, 1, 0, 1], [1, 0, 1, 1]; [1, 0, 0, 0, 1]
---------
n = 6: [5, 1], [1, 5]; [4, 0, 1], [1, 0, 4], [3, 0, 2], [2, 0, 3], [3, 1, 1], [1, 3, 1], [1, 1, 3], [2, 2, 1], [2, 1, 2], [1, 2, 2]; [3, 0, 0, 1], [1, 0, 0, 3], [2, 1, 0, 1], [2, 0, 1, 1], [1, 2, 0, 1], [1, 0, 2, 1], [1, 1, 0, 2], [1, 0, 1, 2], [1, 1, 1, 1]; [2, 0, 0, 0, 1], [1, 0, 0, 0, 2], [1, 1, 0, 0, 1], [1, 0, 1, 0, 1], [1, 0, 0, 1, 1]
---------
n = 7: [6, 1], [1, 6], [5, 2], [2, 5], [4, 3], [3, 4]; [5, 0, 1], [1, 0, 5], [4, 1, 1], [1, 4, 1], [1, 1, 4], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 3, 2], [1, 2, 3]; [4, 0, 0, 1], [1, 0, 0, 4], [3, 0, 0, 2], [2, 0, 0, 3], [3, 1, 0, 1], [3, 0, 1, 1], [1, 3, 0, 1], [1, 0, 3, 1], [1, 1, 0, 3], [1, 0, 1, 3], [2, 2, 0, 1], [2, 0, 2, 1], [2, 1, 0, 2], [2, 0, 1, 2], [1, 2, 0, 2], [1, 0, 2, 2], [2, 1, 1, 1], [1, 2, 1, 1], [1, 1, 2, 1], [1, 1, 1, 2]; [3, 0, 0, 0, 1], [1, 0, 0, 0, 3], [2, 1, 0, 0, 1], [2, 0, 1, 0, 1], [2, 0, 0, 1, 1], [1, 2, 0, 0, 1], [1, 0, 2, 0, 1], [1, 0, 0, 2, 1], [1, 1, 0, 0, 2], [1, 0, 1, 0, 2], [1, 0, 0, 1, 2], [1, 1, 1, 0, 1], [1, 1, 0, 1, 1], [1, 0, 1, 1, 1]; [2, 0, 0, 0, 0, 1],  [1, 0, 0, 0, 0, 2], [1, 1, 0, 0, 0, 1], [1, 0, 1, 0, 0, 1], [1, 0, 0, 1, 0, 1], [1, 0, 0, 0, 1, 1]
-------
x^6 + 1 = (x^2 + 1)*(x^4 - x^2 + 1), hence no [1, 0, 0, 0, 0, 0, 1] recorded, because x^6 - 1 also factorizes.
...
---------------------------------------------------------------------------
Polynomials: n = 4, degree k = 1:  3*x + 1, x + 3; k = 2: 2*x^2 + 1, x^2 + 2, x^2 + x + 1; k = 3: no entry [1, 0, 0, 1], because x^3 + 1 factorizes, as well as x^3 - 1.
---------------------------------------------------------------------------
Height n = 4, degree k = 2, with signed polynomials:
[2, 0, 1] for 2*x^2 + 1, 2*x^2 - 1, [1, 0, 2] for x^2 + 2, x^2 - 2, and [1, 1, 1] for x^2 + x + 1, x^2 + x - 1, x^2 - x + 1, x^2 - x - 1. The corresponding real algebraic numbers come in signed pairs only from 2*x^2 - 1, x^2 - 2, x^2 + x - 1, and x^2 - x - 1, namely, -sqrt(1/2), +sqrt(1/2), -sqrt(2), +sqrt(2), -phi = -A001622, phi - 1, and -(phi - 1), phi. So Cantor's phi (our Phi) is Phi(4, 2) = 8. Together with the four real k = 1 roots from the signed polynomials for [3, 1] and [1, 3] one finds Phi(4) = 12. See A362364.

Georg Cantor, Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen, Journal f. d. reine u. angew. Math. 77 (1874), 258-262.
Georg Cantor, On a Property of the Class of all Real Algebraic Numbers, (English version).
Wolfdieter Lang, Cantor's List of Real Algebraic Numbers of Heights 1 to 7, arXiv:2307.10645 [math.NT], 2023.

Cf. A000051, A001227, A001622, A005409, A364313, A364314, A364315, A364316.

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.

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