A385449 Irregular triangle, read by rows: row n gives the pair of proper positive fundamental solutions (x, y) of the form x^2 - 2*y^2 representing -A057126(n).

1, 1, 4, 3, 1, 2, 5, 4, 2, 3, 6, 5, 1, 3, 9, 7, 3, 4, 7, 6, 1, 4, 13, 10, 4, 5, 8, 7, 3, 5, 11, 9, 2, 5, 14, 11, 5, 6, 9, 8, 1, 5, 17, 13, 6, 7, 10, 9, 1, 6, 21, 16, 5, 7, 13, 11, 7, 8, 11, 10, 4, 7, 16, 13, 3, 7, 19, 15, 2, 7, 22, 17, 1, 7, 25, 19, 8, 9, 12, 11, 5, 8, 17, 14, 7, 9, 15, 13, 3, 8, 23, 18, 9, 10, 13, 12

Offset: 1

Author

Wolfdieter Lang, Jul 11 2025

Keywords

• nonn
• tabf

Comments

The number of (x, y) pairs in row n is 1 for n = 1 and 2, and 2^P, with P = P1 + P7, where P1 and P7 are the number of prime factors 1 modulo 8 and 7 modulo 8, respectively, of A057126(n), for n >= 3.

See A057126 for comments concerning its representation by x^2 - 2*y^2.

The numbers A057126 are given by 2^e_2 * Product_{i=1..P1} p_{1,i}^e_{1,i} * Product_{j=1..P7} p_{7,j}^e_{7,j}, with the odd primes p_{1,i} and p_{7,j} congruent to 1 and 7 modulo 8, respectively. See A007519 and A007522 for these odd primes. Together with 2 these primes are given in A038873, and without 2 in A001132. The exponents are e_2 = 0 or 1, and e_{1,i} and e_{7,j} are nonnegative. The a(1) = 1 is obtained if all exponents vanish. For the proof see Lemma 18 of the linked W. Lang paper, pp. 22 - 23.

The general solutions are obtained from each fundamental solution by application of integer powers of the matrix Auto' = Mat([3,4], [2,3]). See the linked paper eq (28), p. 14, and eq. (40), p. 17 for D = 2, and k = A057126(n). For the explicit form of the powers of Auto' in terms of Chebyshev polynomials S(n, 6) = A001109(n+1) see there eq. (38), and Lemma 10, eq. (43), p. 17.

The conversion to the pair of proper solutions (X, Y) of X^2 - 2*Y^2 = A057126(n) is given by (X, Y) = (2*y - x, x - y). This may result in solutions with negative Y values. They are then transformed to the fundamental positive proper solutions via the mentioned matrix Auto'. See the right part of the example below. For this conversion see also the Nov 09 2009 comment in A035251 by Franklin T. Adams-Watters.

Examples

			n, A057126(n) /k  1  2   3  4 ...   2^P | (X, Y) = (2*y - x, x - y)
-------------------------------------------------------------------
1,  1           | 1  1               1  |  1   0 (3   2)
2,  2           | 4  3               1  |  2   1
3,  7           | 1  2,  5  4        2  |  3  -1 (5   3),  3  1
4, 14 = 2*7     | 2  3,  6  5        2  |  4  -1 (8   5),  4  1
5, 17           | 1  3,  9  7        2  |  5  -2 (7   4),  5  2
6, 23           | 3  4,  7  6        2  |  5  -1 (11  7),  5, 1
7, 31           | 1  4, 13 10        2  |  7  -3 (9   5),  7  3
8, 34 = 2*17    | 4  5,  8  7        2  |  6  -1 (14  9),  6  1
9, 41           | 3  5, 11  9        2  |  7  -2 (13  8),  7  2
10, 46 = 2*23   | 2  5, 14 11        2  |  8  -3 (12  7),  8  3
11, 47          | 5  6,  9  8        2  |  7  -1 (17 11),  7  1
12, 49 = 7^2    | 1  5, 17 13        2  |  9  -4 (11  6),  9  4
13, 62 = 2*31   | 6  7, 10  9        2  |  8  -1 (20 13),  8  1
14, 71          | 1  6, 21 16        2  | 11  -5 (13  7), 11  5
15, 73          | 5  7, 13 11        2  |  9  -2 (19 12),  9  2
16, 79          | 7  8, 11 10        2  |  9  -1 (23 15),  9  1
17, 82 = 2*41   | 4  7, 16 13        2  | 10  -3 (18 11), 10  3
18, 89          | 3  7, 19 15        2  | 11  -4 (17 10), 11  4
19, 94 = 2*47   | 2  7, 22 17        2  | 12  -5 (16  9), 12  5
20, 97          | 1  7, 25 19        2  | 13  -6 (15  8), 13  6
21, 98 = 2*7^2  | 8  9, 12 11        2  | 10  -1 (26 17), 10  1
...
The corresponding fundamental positive proper solutions of X^2 - 2*Y^2 = +119 are: [13 -5 (19 11), 13, 5] and [11 -1 (29 19), 11 1].
		

Links

• Wolfdieter Lang, On Positive Integer Descartes-Steiner Curvature Quintuplets, arXiv:2503.08631 [nucl-th], 2025.

Crossrefs

Cf. A001109, A007519, A007522, A035251, A057126, A038873.