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 A278147 #12 Nov 23 2016 12:56:02 %S A278147 8,0,18,19,0,32,0,33,0,50,34,0,0,0,72,0,52,0,73,0,98,53,0,74,0,99,0, %T A278147 128,0,75,0,100,0,0,0,162,76,0,101,0,0,0,163,0,200,0,102,0,131,0,164, %U A278147 0,201,0,242,103,0,0,0,165,0,202,0,0,0,288,0,133,0,166,0,203,0,244,0,289,0,338,134,0,167,0,204 %N A278147 Triangle read by rows of Cantor pairing function value determining primitive Pythagorean triangles or 0 if there is no such triangle. %C A278147 This entry is inspired by the increasingly ordered nonvanishing entries given in A277557. %C A278147 A primitive Pythagorean triangle is characterized by the pair [n,m], 1 <= m < n, GCD(n,m) = 1 and n+m is odd. The present triangle gives the values T(n, m) = Cantor(m,n) where Cantor(x,y) = (x+y)*(x+y+1)/2 + y. See A277557, also for links. %C A278147 Because the Cantor pairing function N x N -> N is bijective (N = positive integers), all nonzero entries of this triangle appear only once, but here not all positive integers appear. %C A278147 Note that in this triangle in each row the nonvanishing entries increase, but in the first rows up to some n not all T(n, m) values smaller than T(n,n-1) are covered. %C A278147 For the area values of primitive Pythagorean triangles see the table A249869 also for comments on these triangles and references. %F A278147 T(n, m) = (m+n)*(m+n+1)/2 + n, n >= 2, m = 1, 2, ..., n-1, and 0 if GCD(n,m) > 1 or n+m is even. %e A278147 The triangle begins: %e A278147 n\m 1 2 3 4 5 6 7 8 9 10... %e A278147 2: 8 %e A278147 3: 0 18 %e A278147 4: 19 0 32 %e A278147 5: 0 33 0 50 %e A278147 6: 34 0 0 0 7272 %e A278147 7: 0 52 0 73 0 98 %e A278147 8: 53 0 74 0 99 0 128 %e A278147 9: 0 75 0 100 0 0 0 162 %e A278147 10: 76 0 101 0 0 0 163 0 200 %e A278147 11: 0 102 0 131 0 164 0 201 0 242 %e A278147 ... %e A278147 n = 12: 103 0 0 0 165 0 202 0 0 0 288, %e A278147 n = 13: 0 133 0 166 0 203 0 244 0 289 0 338, %e A278147 n = 14: 134 0 167 0 204 0 0 0 290 0 339 0 392, %e A278147 n = 15: 0 168 0 205 0 0 0 291 0 0 0 0 0 450. %e A278147 ... %e A278147 T(3,1) = 0 because 3+1 =4 is even. %e A278147 T(4,2) = 0 because GCD(4,2) = 2 > 1. %e A278147 T(3,2) = (2+3)*(2+3)/2 + 3 = 5*3 + 3 = 18. %e A278147 ... %e A278147 In order to reach all values T(n,m) <= 50 one has to take rows n = 2..6. %e A278147 ... %Y A278147 Cf. A277557, A249869. %K A278147 nonn,tabl,easy %O A278147 2,1 %A A278147 _Wolfdieter Lang_, Nov 21 2016