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 A278147 Triangle read by rows of Cantor pairing function value determining primitive Pythagorean triangles or 0 if there is no such triangle. 1
 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, 128, 0, 75, 0, 100, 0, 0, 0, 162, 76, 0, 101, 0, 0, 0, 163, 0, 200, 0, 102, 0, 131, 0, 164, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 2,1 COMMENTS This entry is inspired by the increasingly ordered nonvanishing entries given in A277557. 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. 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. 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. For the area values of primitive Pythagorean triangles see the table A249869 also for comments on these triangles and references. LINKS FORMULA 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. EXAMPLE The triangle begins: n\m  1   2   3   4   5   6   7   8   9  10... 2:  8 3:  0   18 4:  19   0  32 5:   0  33   0  50 6:  34   0   0   0  7272 7:   0  52   0  73   0  98 8:  53   0  74   0  99   0 128 9:   0  75   0 100   0   0   0 162 10: 76   0 101   0   0   0 163   0 200 11:  0 102   0 131   0 164   0 201   0 242 ... n = 12: 103 0 0 0 165 0 202 0 0 0 288, n = 13: 0 133 0 166 0 203 0 244 0 289 0 338, n = 14: 134 0 167 0 204 0 0 0 290 0 339 0 392, n = 15: 0 168 0 205 0 0 0 291 0 0 0 0 0 450. ... T(3,1) = 0 because 3+1 =4 is even. T(4,2) = 0 because GCD(4,2) = 2 > 1. T(3,2) = (2+3)*(2+3)/2 + 3 = 5*3 + 3 = 18. ... In order to reach all values T(n,m) <= 50 one has to take rows n = 2..6. ... CROSSREFS Cf. A277557, A249869. Sequence in context: A255680 A265115 A214205 * A250109 A022900 A028652 Adjacent sequences:  A278144 A278145 A278146 * A278148 A278149 A278150 KEYWORD nonn,tabl,easy AUTHOR Wolfdieter Lang, Nov 21 2016 STATUS approved

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Last modified March 30 16:16 EDT 2020. Contains 333127 sequences. (Running on oeis4.)