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 A208853 #61 Feb 16 2025 08:33:16 %S A208853 5,13,17,29,25,37,53,41,0,65,85,65,61,73,101,125,97,85,89,109,145,173, %T A208853 137,0,113,0,0,197,229,185,157,145,149,169,205,257,293,241,205,185, %U A208853 181,193,221,265,325,365,305,0,233,221,0 %N A208853 Array of hypotenuses of primitive Pythagorean triangles when read by SW-NE diagonals. %C A208853 All primitive Pythagorean triples (see the links) (a,b,c), with a odd, b even, hence c odd, are given by c=u^2 + v^2, with u odd, u=2*n+1, n>=1, v even, v=2*m, m>=1, and gcd(u,v)=1. The present array is c=c(n,m) = (2*n-1)^2 + (2*m)^2, if gcd(2*n-1,2*m)=1 and 0 otherwise. The corresponding triangle, read by SW-NE diagonals, is T(n,m):= c(n-m+1,m). The 0 entries indicate that there are only non-primitive triples for these n,m values. See the example section for the scaling factor g=gcd(u,v)^2 for such non-primitive triangles. %C A208853 For the increasingly ordered c-values see A008846 (with multiplicity see A020882). %C A208853 All primitive Pythagorean triples are given by %C A208853 (a(n,m)=A208854(n,m), b(n,m)=A208855(n,m), c(n,m)), n>=1, m>=1. If this is (0,0,0) then no primitive triple exists for these n,m values. See the example section. %C A208853 In the prime factorization of c(n,m) (which is odd) all prime factors are of the type 4*k+1 (see A002144). See the Niven-Zuckerman-Montgomery reference, Theorem 3.20, p. 164. For the general representation of positive integers as the sum of two squares see Theorem 2.15 by Fermat, p. 55. E.g.: c(5,2) = 85 = 5*17. c = 5*7^2 = 245 has a non-primitive solution 7^2*(1^2 + 2^2) = 7^2*c(1,1), therefore c(4,7)=0 in this array. %C A208853 The triples with an even cathetus (b) and the hypotenuse (c) differing by 1 unit are (2*k+1, 4*T(k), 4*T(k)+1), k >= 1, with the triangular numbers A000217. The c values are given in A001844. E.g., (n,m)=(1,1), k=1. (3,4,5); (n,m)=(2,1), k=2, (5,12,13); (n,m)=(2,2), k=3, (7,24,25). See the example section for the table. %C A208853 The triples with an odd cathetus (a) and the hypotenuse differing by 2 units are (4*k^2-1, 4*k, 4*k^2+1), k >= 1. These triples are given in (A000466(k), A008586(k), A053755(k)). E.g., (n,m)=(1,4), k=4, (63,16,65). %C A208853 The triples with the catheti differing by one length unit are generated by a substitution rule for the (u,v) values starting with (1,1). See a _Wolfdieter Lang_ comment on A001653 for this rule. - _Wolfdieter Lang_, Mar 08 2012 %D A208853 I. Niven, H. S. Zuckerman and H.L. Montgomery, An Introduction to the Theory of Numbers, 5th edition, Wiley & Sons, New York, 1991 %H A208853 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html">Pythagorean Triples and Online Calculators</a>. %H A208853 E. S. Rowland, <a href="https://ericrowland.github.io/investigations/tripleslist-long.html">Primitive Solutions to x^2 + y^2 = z^2</a> %H A208853 Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/rtritab.txt">Table of primitive Pythagorean triples and related parameters</a> %H A208853 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PythagoreanTriple.html">Pythagorean Triple</a>. %F A208853 T(n,m) = c(n-m+1,m), n >= m >= 1, with c(n,m) := (2*n-1)^2 + (2*m)^2, if gcd(2*n-1, 2*m) = 1 and 0 otherwise. %e A208853 Triangle T(n,m): %e A208853 ......m| 1 2 3 4 5 6 7 8 9 10 ... %e A208853 ......v| 2 4 6 8 10 12 14 16 18 20 ... %e A208853 n, u %e A208853 1, 1 5 %e A208853 2, 3 13 17 %e A208853 3, 5 29 25 37 %e A208853 4, 7 53 41 0 65 %e A208853 5, 9 85 65 61 73 101 %e A208853 6, 11 125 97 85 89 109 145 %e A208853 7, 13 173 137 0 113 0 0 197 %e A208853 8, 15 229 185 157 145 149 169 205 257 %e A208853 9, 17 293 241 205 185 181 193 221 265 325 %e A208853 10,19 365 305 0 233 221 0 0 281 0 401 %e A208853 ... %e A208853 Array c(n,m): %e A208853 ......m| 1 2 3 4 5 6 7 8 9 10 ... %e A208853 ......v| 2 4 6 8 10 12 14 16 18 20 ... %e A208853 n, u %e A208853 1, 1 5 17 37 65 101 145 197 257 325 401 %e A208853 2 3 13 25 0 73 109 0 205 265 0 409 %e A208853 3, 5 29 41 61 89 0 169 221 281 349 0 %e A208853 4, 7 53 65 85 113 149 193 0 305 373 449 %e A208853 5, 9 85 97 0 145 181 0 277 337 0 481 %e A208853 6, 11 125 137 157 185 221 265 317 377 445 521 %e A208853 7, 13 173 185 205 233 269 313 365 425 493 569 %e A208853 8, 15 229 241 0 289 0 0 421 481 0 0 %e A208853 9, 17 293 305 325 353 389 433 485 545 613 689 %e A208853 10,19 365 377 397 425 461 505 557 617 685 761 %e A208853 ... %e A208853 ------------------------------------------------------------------ %e A208853 Array of triples (a(n,m)=A208854,b(n,m)=A208855,c(n,m)): %e A208853 ......m| 1 2 3 4 ... %e A208853 ......v| 2 4 6 8 ... %e A208853 n, u %e A208853 1, 1 (3,4,5) (15,8,17) (35,12,37) (63,16,65) %e A208853 2, 3 (5,12,13) (7,24,25) (0,0,0) (55,48,73) %e A208853 3, 5 (21,20,29) (9,40,41) (11,60,61) (39,80,89) %e A208853 4, 7 (45,28,53) (33,56,65) (13,84,85) (15,112,113) %e A208853 5, 9 (77,36,85) (65,72,97) (0,0,0) (17,144,145) %e A208853 6, 11 (117,44,125) (105,88,137) (85,132,157) (57,176,185) %e A208853 7, 13 (165,52,173) (153,104,185) (133,156,205) (105,208,233) %e A208853 8, 15 (221,60,229) (209,120,241) (0,0,0) (161,240,289) %e A208853 9, 17 (285,68,293) (273,136,305) (253,204,325) (225,272,353) %e A208853 10,19 (357,76,365) (345,152,377) (325,228,397) (297,304,425) %e A208853 ... %e A208853 Array continued: %e A208853 Array of triples (a(n,m)=A208854,b(n,m)=A208855,c(n,m)): %e A208853 ......m| 5 6 7 8 ... %e A208853 ......v| 10 12 14 16 ... %e A208853 n, u %e A208853 1, 1 (99,20,101) (143,24,145) (195,28,197) (255,32,257) %e A208853 2 3 (91,60,109) (0,0,0) (187,84,205) (247,96,265) %e A208853 3, 5 (0,0,0) (119,120,169) (171,140,221) (231,160,281) %e A208853 4, 7 (51,140,149) (95,168,193) (0,0,0) (207,224,305) %e A208853 5, 9 (19,180,181) (0,0,0) (115,252,277) (175,288,337) %e A208853 6, 11 (21,220,221) (23,264,265) (75,308,317) (135,352,377) %e A208853 7, 13 (69,260,269) (25,312,313) (27,364,365) (87,416,425) %e A208853 8, 15 (0,0,0) (0,0,0) (29,420,421) (31,480,481) %e A208853 9, 17 (189,340,389) (145,408,433) (93,476,485) (33,544,545) %e A208853 10,19 (261,380,461) (217,456,505) (165,532,557) (105,608,617) %e A208853 ... %e A208853 (0,0,0) indicates that no primitive Pythagorean triangle exists for these (n,m) values. The corresponding scaled triples would be (a,b,c) = g*(a/g,b/g,c/g), with g=gcd(u,v)^2 for such non-primitive triangles. E.g., c(n,m) = c(5,3) = 0, (u,v) = (9,6), g = 3^2, (45,108,117) = 3^2*(45/9,108/9,117/9) = 9*(5,12,13). The scaling factor for the primitive triangle (5,12,13), tabulated for c(n,m)=(2,1), is here 9. %Y A208853 Cf. A020882, A002144, A208854, A208855. %K A208853 nonn,easy,tabl %O A208853 1,1 %A A208853 _Wolfdieter Lang_, Mar 05 2012