This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A385449 #9 Jul 27 2025 00:20:50
%S A385449 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,
%T A385449 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,
%U A385449 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
%N 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).
%C A385449 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.
%C A385449 See A057126 for comments concerning its representation by x^2 - 2*y^2.
%C A385449 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.
%C A385449 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.
%C A385449 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.
%H A385449 Wolfdieter Lang, <a href="https://arxiv.org/abs/2503.08631">On Positive Integer Descartes-Steiner Curvature Quintuplets</a>, arXiv:2503.08631 [nucl-th], 2025.
%e A385449 n, A057126(n) /k 1 2 3 4 ... 2^P | (X, Y) = (2*y - x, x - y)
%e A385449 -------------------------------------------------------------------
%e A385449 1, 1 | 1 1 1 | 1 0 (3 2)
%e A385449 2, 2 | 4 3 1 | 2 1
%e A385449 3, 7 | 1 2, 5 4 2 | 3 -1 (5 3), 3 1
%e A385449 4, 14 = 2*7 | 2 3, 6 5 2 | 4 -1 (8 5), 4 1
%e A385449 5, 17 | 1 3, 9 7 2 | 5 -2 (7 4), 5 2
%e A385449 6, 23 | 3 4, 7 6 2 | 5 -1 (11 7), 5, 1
%e A385449 7, 31 | 1 4, 13 10 2 | 7 -3 (9 5), 7 3
%e A385449 8, 34 = 2*17 | 4 5, 8 7 2 | 6 -1 (14 9), 6 1
%e A385449 9, 41 | 3 5, 11 9 2 | 7 -2 (13 8), 7 2
%e A385449 10, 46 = 2*23 | 2 5, 14 11 2 | 8 -3 (12 7), 8 3
%e A385449 11, 47 | 5 6, 9 8 2 | 7 -1 (17 11), 7 1
%e A385449 12, 49 = 7^2 | 1 5, 17 13 2 | 9 -4 (11 6), 9 4
%e A385449 13, 62 = 2*31 | 6 7, 10 9 2 | 8 -1 (20 13), 8 1
%e A385449 14, 71 | 1 6, 21 16 2 | 11 -5 (13 7), 11 5
%e A385449 15, 73 | 5 7, 13 11 2 | 9 -2 (19 12), 9 2
%e A385449 16, 79 | 7 8, 11 10 2 | 9 -1 (23 15), 9 1
%e A385449 17, 82 = 2*41 | 4 7, 16 13 2 | 10 -3 (18 11), 10 3
%e A385449 18, 89 | 3 7, 19 15 2 | 11 -4 (17 10), 11 4
%e A385449 19, 94 = 2*47 | 2 7, 22 17 2 | 12 -5 (16 9), 12 5
%e A385449 20, 97 | 1 7, 25 19 2 | 13 -6 (15 8), 13 6
%e A385449 21, 98 = 2*7^2 | 8 9, 12 11 2 | 10 -1 (26 17), 10 1
%e A385449 ...
%e A385449 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].
%Y A385449 Cf. A001109, A007519, A007522, A035251, A057126, A038873.
%K A385449 nonn,tabf
%O A385449 1,3
%A A385449 _Wolfdieter Lang_, Jul 11 2025