A308686 Irregular triangle with the nonnegative proper fundamental solutions of the binary quadratic form x^2 + x*y - y^2 representing N = N(n) = A089270(n), for n >= 1.

1, 0, 2, 1, 3, 1, 3, 2, 4, 1, 4, 3, 5, 1, 5, 4, 5, 2, 5, 3, 6, 1, 6, 5, 7, 1, 7, 6, 7, 2, 7, 5, 7, 3, 7, 4, 8, 1, 8, 7, 8, 3, 8, 5, 9, 1, 9, 8, 9, 2, 9, 7, 9, 4, 9, 5, 10, 1, 10, 9, 10, 3, 10, 7, 11, 1, 11, 10, 11, 2, 11, 9, 11, 3, 11, 8, 11, 4, 11, 7, 11, 5, 11, 6, 12, 1, 12, 11, 12, 5, 12, 7, 13, 1, 13, 12, 13, 2, 13, 11, 13, 3, 13, 10, 13, 4, 13, 9, 13, 5, 13, 8, 14, 1, 14, 13, 13, 6, 13, 7

Offset

1,3

Comments

The length of row n is 2 for n = 1, 2; 4 for n = 3..28, 30..40, 42, 44..58, 60...; 8 for 29, 41, 43, 59,...; 16 for 643, 688, 896, ...; ... .

The numbers N with row length 8 are 209, 319, 341, 451, 551, 589, 649, 671, 779, 781, 869, 899, 979, 1045, 1111, ...; with row length 16 they are 6061, 6479, 8569, 9889, ...; .... .

The fundamental solution (x, y) with gcd(x, y) = 1 (proper solutions) are listed pairwise for n >= 3 (N >= 11) and enclosed in square brackets in the example, Within a square bracket the numbers y always sum to x.

For the numbers N with a solution (x, 1) see A028387(n-1), for n >= 1. There N = 1 is included by taking the solution (1, 1) instead of (1, 0).

The general solutions are then obtained by applying integer powers of the automorphic matrix Auto(50) = Matrix([1, 1],[1, 2]) on these fundamental solutions. The matrix Auto(5) is related to the 2-cycle of the principal reduced form F_p = [1, 1, -1] and the reduced form F' = [-1, 1, 1].

See the W. Lang link in A089270 for proofs and Tables. Here Table 4.

Links

Table of n, a(n) for n=1..120.

Example

The irregular triangle T(n, k) begins (the solutions are (x, y)):
n,    N \ k  1  2    3   4       5  6    7   8    ...
1,    1:    (1  0) [sometimes (1, 1)]
2,    5:    (2  1)
3,   11:   [(3  1)  (3   2)]
4,   19:   [(4  1)  (4   3)]
5,   29:   [(5  1)  (5   4)]
6,   31:   [(5  2)  (5   3)]
7,   41:   [(6  1)  (6   5)]
8,   55:   [(7  1)  (7   6)]
9,   59:   [(7  2)  (7   5)]
10,  61:   [(7  3)  (7   4)]
11,  71:   [(8  1)  (8   7)]
12,  79:   [(8  3)  (8   5)]
13,  89:   [(9  1)  (9   8)]
14,  95:   [(9  2)  (9   7)]
15, 101:   [(9  4)  (9   5)]
16, 109:  [(10  1) (10   9)]
17, 121:  [(10  3) (10   7)]
18, 131:  [(11  1) (11  10)]
19, 139:  [(11  2) (11   9)]
20, 145:  [(11  3) (11   8)]
...
29, 209:  [(13  5) (13   8)]  [(14  1) (14  13)]
30, 211:  [(13  6) (13   7)]
...

Crossrefs

Cf. A028387, A089270.

Sequence in context: A316436 A303674 A038569 * A020650 A361373 A124224

Adjacent sequences: A308683 A308684 A308685 * A308687 A308688 A308689

Keyword

nonn,tabf

Author

Wolfdieter Lang, Jul 05 2019

Status

approved