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A336888
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Irregular triangle read by rows: row n gives the pairs x(n)_j, y(n)_j, for j = 1, 2, ..., A336889(n), such that the legs of nonrectangular (nondegenerate) triangles X(n)_j = sqrt(x(n)_j), Y(n)_j = sqrt(y(n)_j), with base Z(n) = sqrt(z(n)), and integers 1 <= x(n)_j <= y(n)_j <= z(n), lead to primitive quartets (x(n)_j, y(n)_j, z(n), A(n)_j) with positive integer area A(n)_j. Hence z(n) = A337215(n) and A(n)_j = A337216(n, j).
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3
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4, 5, 1, 5, 5, 8, 2, 4, 4, 10, 4, 5, 1, 8, 5, 5, 5, 13, 13, 13, 4, 5, 5, 8, 8, 13, 4, 17, 4, 10, 5, 9, 2, 10, 1, 13, 5, 13, 1, 17, 9, 17, 13, 17, 5, 8, 4, 13, 8, 13, 4, 17, 16, 17, 1, 20, 4, 10, 2, 20, 10, 20, 18, 20, 4, 26, 8, 9, 4, 13, 5, 16, 1, 20, 13, 20, 17, 20, 8, 25, 16, 29
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OFFSET
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1,1
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COMMENTS
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The length of row n is 2*A336889(n).
Row n >= 1 of this irregular triangle is obtained from row n of A336885 after elimination of all pairs x(n)_j = A336885(n, 2*j-1), y(n)_j = A336885(n, 2*j), for j = 1, 2, ..., A336886(n), which lead to imprimitive quartets (x(n)_j, y(n)_j, z(n), A(n)_j) with z(n) = A334818(n) and A(n)_j = A336887(n, j).
Note that the present triples (x(n)_j, y(n)_j, z(n)) with x(n)_j = a(n, 2*j - 1), y(n)_j = a(n, 2*j), for j = 1, 2, ..., A336889(n), and z(n) = A337215(n), are not always primitive. E.g., (a(4, 1), a(4, 2)) = (2, 4) with z(4) = 10 survived the purgation of row n = 4 of A336885 because the area is A_1(4) = A336887(4, 1) = 1 = A337216(4, 1). Similarly (4, 10) with area 3 survived in this row, but the pair (8, 10) with area 4 has been eliminated because it doubles the quartet (4, 5, 5, 2) obtained from row n = 1.
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LINKS
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FORMULA
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Bisection of index k: With a(n, 2*j-1) = x(n)_j and a(n, 2*j) = y(n)_j, and z(n) = A337215(n), where 1 <= x(n)_j <= y(n)_j <= z(n), the nonrectangular triangle with sides (sqrt(x(n)_j), sqrt(y(n)_j), sqrt(z(n))) has positive integer area A(n)_j = A337216(n, j), and the quartet (x(n)_j, y(n)_j, z(n), A(n)_j) is primitive, for j = 1, 2, ..., A336889(n), and n >= 1.
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EXAMPLE
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The irregular triangle a(n, k) begins (z(n) = A337215(n)):
n, z(n) \ k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 ...
---------------------------------------------------------------------------
1, 5: 4 5
2, 8: 1 5
3, 9: 5 8
4, 10: 2 4 4 10
5, 13: 4 5 1 8
6, 16: 5 5 5 13 13 13
7, 17: 4 5 5 8 8 13 4 17
8, 18: 4 10
9, 20: 5 9 2 10 1 13 5 13 1 17 9 17 13 17
10, 25: 5 8 4 13 8 13 4 17 16 17 1 20
11, 26: 4 10 2 20 10 20 18 20 4 26
12, 29: 8 9 4 13 5 16 1 20 13 20 17 20 8 25 16 29
13, 32: 5 13 9 17 1 25 17 25 5 29
14, 34: 4 18 2 20 10 20 4 26 20 26
15, 36: 10 10 13 13 5 17 13 25 25 25 2 26 5 29 10 34 34 34
16: 37: 5 16 4 17 8 17 5 20 13 20 8 25 20 25 13 32 29 32 4 37
17, 40: 9 13 5 17 13 17 13 25 1 29 17 29 25 29 1 37 9 37
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n = 18, z(n) = 41: 8 13 16 17 5 20 9 20 4 25 4 29 1 32 17 36 29 36 20 37
5 40 13 40 37 40;
n = 19, z(n) = 45: 13 16 8 17 17 20 4 25 20 29 5 32 4 37 16 37 1 40 8 41;
n = 20, z(n) = 49: 13 20 8 29 25 32 5 40 20 41;
n = 21, z(n) = 50: 18 20 4 26 4 34 20 34 2 36 26 36 4 50.
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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