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Irregular triangle read by rows: a(n, j) gives the positive integer area A(n)_j corresponding to the nonrectangular triangles with sides (sqrt(x(n)_j), sqrt(y(n)_j), z(n)), with integers 1 <= x(n)_j <= y(n)_j <= z(n), that lead to primitive quartets (x(n)_j, y(n)_j, z(n), a(n, j)). Hence x(n)_j = A336888(n, 2*j-1), y(n)_j = A336888(n, 2*j), for j = 1, 2, ..., A336889(n), and z(n) = A337215(n), for n >= 1.
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%I #8 Aug 25 2020 05:49:01

%S 2,1,3,1,3,2,1,2,4,6,1,3,5,4,3,3,1,1,4,2,6,7,1,3,5,4,8,2,1,3,7,9,5,3,

%T 2,4,1,8,9,7,10,2,6,2,10,6,3,1,7,5,11,3,6,3,9,12,3,6,9,15,2,1,5,4,8,7,

%U 11,10,14,6,3,1,7,9,1,11,13,3,9,1,8,3,6,4,5,2,12,15,13,7,11,17

%N Irregular triangle read by rows: a(n, j) gives the positive integer area A(n)_j corresponding to the nonrectangular triangles with sides (sqrt(x(n)_j), sqrt(y(n)_j), z(n)), with integers 1 <= x(n)_j <= y(n)_j <= z(n), that lead to primitive quartets (x(n)_j, y(n)_j, z(n), a(n, j)). Hence x(n)_j = A336888(n, 2*j-1), y(n)_j = A336888(n, 2*j), for j = 1, 2, ..., A336889(n), and z(n) = A337215(n), for n >= 1.

%F a(n, j) = (1/4)*sqrt(2*(z(n)*y(n)_j + z(n)*x(n)_j + y(n)_j*x(n)_j) - ((x(n)_j)^2 + (y(n)_j)^2 + z(n)^2)), for j = 1, 2, ..., A336889(n), with x(n)_j = A336888(n, 2*j-1), y(n)_j = A336888(n, 2*j) and z(n) = A337215(n), for n >= 1.

%e The irregular triangle a(n,j) begins (z(n) = A337215(n)):

%e n, z(n) \ j 1 2 3 4 5 6 7 8 9 10 11 12 13 ...

%e -----------------------------------------------------

%e 1, 5: 2

%e 2, 8: 1

%e 3, 9: 3

%e 4, 10: 1 3

%e 5, 13: 2 1

%e 6, 16: 2 4 6

%e 7, 17: 1 3 5 4

%e 8, 18: 3

%e 9, 20: 3 1 1 4 2 6 7

%e 10, 25: 1 3 5 4 8 2

%e 11, 26: 1 3 7 9 5

%e 12, 29: 3 2 4 1 8 9 7 10

%e 13, 32: 2 6 2 10 6

%e 14, 34: 3 1 7 5 11

%e 15, 36: 3 6 3 9 12 3 6 9 15

%e 16: 37: 2 1 5 4 8 7 11 10 14 6

%e 17, 40: 3 1 7 9 1 11 13 3 9

%e 18, 41: 1 8 3 6 4 5 2 12 15 13 7 11 17

%e 19, 45: 6 3 9 3 12 6 6 12 3 9

%e 20, 49: 7 7 14 7 14

%e 21, 50: 9 1 5 13 3 15 7

%e ...

%e -----------------------------------------------------

%e a(6, 3) = 6 for the triangle from row n = 6 of A336888: (x(6)_5 , y(6)_6, z(6)) = (13, 13, 16), with area (in some square length units) (1/4)*sqrt(2*(16*13 + 16*13 + 13*13) - (2*13^2 +16^2)) = 6.

%Y Cf. A334818, A336885, A336885, A336887, A336889 (row lengths), A337215 (z(n)), A337216 (areas).

%K nonn,tabf

%O 1,1

%A _Wolfdieter Lang_, Aug 19 2020