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 A338275 #38 Jan 05 2022 10:50:57 %S A338275 5,8,9,5,12,13,15,12,19,5,16,17,21,20,29,9,24,25,5,20,21,27,28,39,9, %T A338275 40,41,29,16,33,5,24,25,33,36,49,9,56,57,45,28,53,23,32,39,5,28,29,39, %U A338275 44,59,9,72,73,61,40,73,33,56,65,13,48,49 %N A338275 Array of triples read by antidiagonals of triples, giving analogs of Pythagorean triples [a,b,c] being [odd,even,odd] consistent with the functions U(i;n,k) described in A327263. %C A338275 In forming the triples we follow what we know about calculating Pythagorean triples given two positive integers m > n. That is, a = m^2 - n^2; b = 2m*n; c = m^2 + n^2. This is the case when i = 2. Here m is odd and n is even. The rows of triples are sorted by m then by n. %C A338275 Within all rows of triples, each of a, b, and c are arithmetic progressions. %C A338275 Within all rows of triples, consecutive triples have the same difference (delta) which is always an even multiple of a primitive Pythagorean triple. %C A338275 delta_a = ((m-1)^2 - n^2)/2, %C A338275 delta_b = (m-1)*n, %C A338275 delta_c = ((m-1)^2 + n^2)/2. %C A338275 When n = m - 1, delta_a = 0 while delta_b = delta_c, so delta_a^2 + delta_b^2 = delta_c^2 trivially. %C A338275 In rows with n = m - 1, the following are true for all i: %C A338275 a = m + n, %C A338275 c = b + 1, %C A338275 b + c = U(i; a, a). %C A338275 Within all columns of triples, for each m, b is an arithmetic progression with difference 2m+2, a has a constant second difference of -4i and c has a constant second difference of +4i. %C A338275 From _David Lovler_, Dec 04 2020: (Start) %C A338275 If we modify the Pythagorean inradius formula according to the rules of U(i;odd,even), r = (m-n)*n becomes r = (i*(m-n)*n - (i-2)*n)/2. To distinguish this from the usual inradius let us call it the inradius computation or irc. The irc might not have a Euclidean interpretation, but using it brings light to the following theorem. Within a row of the table, the inradius of the (constant, Pythagorean) difference between consecutive triples equals the difference between the ircs of consecutive triples in the row, and both equal (m-n-1)*n/2. %C A338275 Proof: The left hand side, according to A338895 and A338896, equals (m-n-1)*n/2. For the right hand side, given odd m and even n, irc(i+1) - irc(i) = U(i+1; m-n, n) - U(i; m-n, n) = ((i+1)*(m-n)*n - (i+1-2)*n)/2 - (i*(m-n)*n - (i-2)*n)/2 = (m-n-1)*n/2. (End) %H A338275 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a> %F A338275 Given i > 0, for each (odd,even) pair (m,n) with m >= 3 and m > n >= 2, the triple [a, b, c] consists of %F A338275 a = (i*m^2 - (i-2)*(2m-1))/2 - (i*n^2)/2 analogous to m^2 - n^2 %F A338275 b = i*m*n - (i-2)*n analogous to 2m*n %F A338275 c = (i*m^2 - (i-2)*(2m-1))/2 + (i*n^2)/2 analogous to m^2 + n^2. %e A338275 In the following start of the array, the column headings would be U(i;n,k), but n and k are left out to avoid confusion with n of (m,n). %e A338275 U(1;,) U(2;,) U(3;,) U(4;,) U(5;,) U(6;,) %e A338275 m 3 %e A338275 n 2 [5,8,9] [5,12,13] [5,16,17] [5,20,21] [5,24,25] [5,28,29] %e A338275 ----------------------------------------------------------------------------------- %e A338275 m 5 %e A338275 n 2 [15,12,19] [21,20,29] [27,28,39] [33,36,49] [39,44,59] [45,52,69] %e A338275 n 4 [9,24,25] [9,40,41] [9,56,57] [9,72,73] [9,88,89] [9,104,105] %e A338275 ----------------------------------------------------------------------------------- %e A338275 m 7 %e A338275 n 2 [29,16,33] [45,28,53] [61,40,73] [77,52,93] [93,64,113] [109,76,133] %e A338275 n 4 [23,32,39] [33,56,65] [43,80,91] [53,104,117] [63,128,143] [73,152,169] %e A338275 n 6 [13,48,49] [13,84,85] [13,120,121] [13,156,157] [13,192,193] [13,228,229] %e A338275 ----------------------------------------------------------------------------------- %e A338275 m 9 %e A338275 n 2 [47,20,51] [77,36,85] [107,52,119] [137,68,153] [167,84,187] [197,100,221] %e A338275 n 4 [41,40,57] [65,72,97] [89,104,137] [113,136,177] [137,168,217] [161,200,257] %e A338275 n 6 [31,60,67] [45,108,117] [59,156,167] [73,204,217] [87,252,267] [101,300,317] %e A338275 n 8 [17,80,81] [17,144,145] [17,208,209] [17,272,273] [17,336,337] [17,400,401] %Y A338275 Cf. A338895, A338896. %K A338275 nonn,tabf %O A338275 1,1 %A A338275 _David Lovler_, Oct 19 2020