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.
FORMULA
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
KEYWORD
nonn,easy,tabf
AUTHOR
Wolfdieter Lang, Apr 08 2021
STATUS
approved