A343232 Irregular triangle T read by rows: T(n, m) gives the solutions j of the congruence A002061(j+1) = j^2 + j + 1 == 0 (mod k(n)), with k(n) = A034017(n+1), for j from {0, 1, ..., k(n)-1}, and n >= 1.

0, 1, 2, 4, 3, 9, 7, 11, 4, 16, 5, 25, 10, 26, 16, 22, 6, 36, 18, 30, 7, 49, 13, 47, 29, 37, 8, 64, 23, 55, 9, 16, 74, 81, 25, 67, 35, 61, 46, 56, 45, 63, 10, 100, 19, 107, 49, 79, 11, 30, 102, 121, 42, 96, 67, 79

Offset

1,3

Comments

The length of row n is A341422(n), the number of representative parallel primitive forms (rpapfs) for positive binary quadratic forms of Discriminant = -3 representing k = k(n) = A034017(n+1), for n >= 1.

These rpapfs for each j are [k(n), 2*j+1, (j^2 + j + 1)/k(n)], for n >= 1.

The solutions for k(n) >= 7 come in pairs j and k(n) - (1 + j). For k(1) = 1 and k(2) = 3 these pairs collapse to one solution.

Links

Table of n, a(n) for n=1..56.

Formula

T(n, m) gives the solutions j of A002061(j+1) = j^2 + j + 1 == 0 (mod k(n)), for k(n) = A034017(n+1), for n >= 1.

Example

The irregular triangle T(n, m) begins:
n,   k(n)\m  1   2   3   4 ...   rpapfs
1,    1:     0                  [1,1,1]
2,    3:     1                  [3,3,1]
3,    7:     2   4              [7,5,1],      [7,9,3]
4,   13:     3   9              [13,7,1],     [13,19,7]
5,   19:     7  11              [19,15,3],    [19,23,7]
6,   21:    14  16              [21,9,1],     [21,33,13]
7,   31:     5  25              [31,11,1],    [31,51,21]
8,   37:    10  26              [37,21,3],    [37,53,19]
9,   39:    16  22              [39,33,7],    [39,45,13]
10,  43:     6  36              [43,13,1],    [43,73,31]
11,  49:    18  30              [49,37,7],    [49,61,19]
12,  57:     7  49              [57,15,1],    [57,99,43]
13,  61:    13  47              [61,27,3],    [61,95,37]
14,  67:    29  37              [67,59,13],   [67,75,21]
15,  73:     8  64              [73,17,1],    [73,129,57]
16,  79:    23  55              [79,47,7],    [79,111,39]
17,  91:     9  16              [91,19,1],    [91,33,3], [91,149,61],
                                [91,163,73]
18,  93:    25  67              [93,51,7],    [93,135,49]
19,  97:    35  61              [97,71,13] ,  [97,123,39]
20, 103:    46  56              [103,93,21],  [103,113,31]
21, 109:    45  63              [109,91,19],  [109,127,37]
22, 111:    10 100              [111,21,1],   [111,201,91]
23, 127:    19 107              [127,39,3],   [127,215,91]
24, 129:    49  79              [129,99,19],  [129,159,49]
25, 133:    11  30 102 121      [133, 23,1],  [133,61,7], [133,205,79],
                                [133,243,111]
26, 139     42  96              [139,85,13],  [139,193, 67]
27, 147:    67  79              [147,135,31], [147,159,43]
28, 151:    32 118              [151,65,7],   [151,237,93]
29, 157:    12 144              [157,25,1],   [157,289,133]
30, 163:    58 104              [163,117,21], [163,209,67]
...

Crossrefs

Cf. A002061, A034017, A341422.

Sequence in context: A329901 A284572 A157182 * A292145 A297552 A286555

Adjacent sequences: A343229 A343230 A343231 * A343233 A343234 A343235

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang, Apr 08 2021

Status

approved