%I #12 Jul 24 2019 02:42:15

%S 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,

%T 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,

%U 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

%N 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.

%C 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, ...; ... .

%C 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, ...; .... .

%C 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.

%C 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).

%C 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].

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

%e The irregular triangle T(n, k) begins (the solutions are (x, y)):

%e n, N \ k 1 2 3 4 5 6 7 8 ...

%e 1, 1: (1 0) [sometimes (1, 1)]

%e 2, 5: (2 1)

%e 3, 11: [(3 1) (3 2)]

%e 4, 19: [(4 1) (4 3)]

%e 5, 29: [(5 1) (5 4)]

%e 6, 31: [(5 2) (5 3)]

%e 7, 41: [(6 1) (6 5)]

%e 8, 55: [(7 1) (7 6)]

%e 9, 59: [(7 2) (7 5)]

%e 10, 61: [(7 3) (7 4)]

%e 11, 71: [(8 1) (8 7)]

%e 12, 79: [(8 3) (8 5)]

%e 13, 89: [(9 1) (9 8)]

%e 14, 95: [(9 2) (9 7)]

%e 15, 101: [(9 4) (9 5)]

%e 16, 109: [(10 1) (10 9)]

%e 17, 121: [(10 3) (10 7)]

%e 18, 131: [(11 1) (11 10)]

%e 19, 139: [(11 2) (11 9)]

%e 20, 145: [(11 3) (11 8)]

%e ...

%e 29, 209: [(13 5) (13 8)] [(14 1) (14 13)]

%e 30, 211: [(13 6) (13 7)]

%e ...

%Y Cf. A028387, A089270.

%K nonn,tabf

%O 1,3

%A _Wolfdieter Lang_, Jul 05 2019