A278717 Triangle read by rows: T(n, m) gives the difference between the even and odd leg of the primitive Pythagorean triangle determined by (n, m) with n > m >= 1, gcd(n, m) = 1 and n+m odd, or 0 for other (n, m).

1, 0, 7, -7, 0, 17, 0, -1, 0, 31, -23, 0, 0, 0, 49, 0, -17, 0, 23, 0, 71, -47, 0, -7, 0, 41, 0, 97, 0, -41, 0, 7, 0, 0, 0, 127, -79, 0, -31, 0, 0, 0, 89, 0, 161, 0, -73, 0, -17, 0, 47, 0, 119, 0, 199, -119, 0, 0, 0, 1, 0, 73, 0, 0, 0, 241, 0, -113, 0, -49, 0, 23, 0, 103, 0, 191, 0, 287, -167, 0

Offset: 2

Author

Wolfdieter Lang, Nov 29 2016

Keywords

• sign
• tabl
• easy

Comments

T(n, m) also gives twice the member r(n, m) of the triple (r(n, m), s(n, m), t(n, m)) with squares r(n, m)^2, s(n, m)^2 and t(n, m)^2 in arithmetic progression with common difference A(n, m) = A249869(n, m), the area of the primitive Pythagorean triangle. The members 2*s(n, m) (hypotenuse) and 2*t(n, m) (sum of catheti) are given in A222946(n, m) and A225949(n, m), respectively.

There is a one-to-one correspondence between rational Pythagorean triangles (a,b,c) with area A and three squares r^2, s^2 and t^2 in arithmetic progression with common difference A > 0: (r, s, t) = ((b-a)/2, c/2, (b+a)/2) and (a,b,c) = (t-r, t+r, 2*s). See the Keith Conrad link, Theorem 3.1. Leg exchange leads to the same progression of squares but r is exchanged with -r.

Here only primitive Pythagorean triangles with even leg b and area A given in A249869 are considered. See A249866, also for references.

Examples

			The triangle T(n, m) begins:
n\m   1   2   3   4  5  6  7   8    9   10
2:    1
3:    0   7
4:   -7   0  17
5:    0  -1   0  31
6:  -23   0   0   0 49
7:    0 -17   0  23  0 71
8:  -47   0  -7   0 41  0 97
9:    0 -41   0   7  0  0  0 127
10: -79   0 -31   0  0  0 89   0  161
11:   0 -73   0 -17  0 47  0 119    0  199
n\m   1   2   3   4  5  6  7   8    9   10
...
n = 12: -119 0 0 0 1 0 73 0 0 0 241,
n = 13: 0 -113 0 -49 0 23 0 103 0 191 0 287,
n = 14: -167 0 -103 0 -31 0 0 0 137 0 233 0 337,
n = 15: 0 -161 0 -89 0 0 0 79 0 0 0 0 0 391.
...
The triples (2*r(n, m),2*s(n, m),2*t(n, m)) begin (we abbreviate here (0,0,0) by 0):
n\m   1           2         3           4 ...
2: (1,5,7)
3:    0       (7,13,17)
4: (-7,17,23)     0     (17,25,31)
5:     0      (-1,29,41)     0      (31,41,49)
...
n = 6: (-23, 37, 47) 0 0 0 (49,61,71),
n = 7:  0 (-17,53,73) 0 (23,65,89) 0 (71,85,97),
n = 8: (-47,65,79) 0 (-7,73,103) 0 (41,89,119) 0 (97,113,127),
n = 9:  0 (-41,85,113) 0 (7,97,137) 0 0 0 (127,145,161),
n =10: (-79, 101, 119) 0 (-31,109,151) 0 0 0 (89,149,191) 0 (161,181,199).
...
The quartets (r(n,m)^2,s(n,m)^2,t(n,m)^2;A(n, m)) of squares in arithmetic progression with common difference A(n,m) = A249869(n,m) begin (here (0,0,0;0) is abbreviated as 0):
n = 2: (1/4,25/4,49/4;6),
n = 3: 0 (49/4,169/4,289/4;30),
n = 4: (49/4,289/4,529/4;60) 0 (289/4,625/4,961/4;84),
n = 5: 0 (1/4,841/4,1681/4;210) 0 (961/4,1681/4,2401/4;180),
n = 6: (529/4,1369/4,2209/4;210) 0 0 0 (2401/4,3721/4,5041/4;330),
n = 7: 0 (289/4,2809/4,5329/4;630) 0 (529/4,4225/4,7921/4;924) 0 (5041/4,7225/4,9409/4;546),
n = 8: (2209/4,4225/4,6241/4;504) 0 (49/4,5329/4,10609/4;1320) 0 (1681/4,7921/4,14161/4;1560) 0, (9409/4,12769/4,16129/4;840),
n = 9: 0, (1681/4,7225/4,12769/4;1386) 0 (49/4,9409/4,18769/4;2340) 0 0 0 (16129/4,21025/4,25921/4;1224)
n = 10: (6241/4,10201/4,14161/4);990) 0 (961/4,11881/4,22801/4;2730) 0 0 0 (7921/4,22201/4,36481/4;3570) 0 (25921/4,32761/4,39601/4;1710).
...
		

Links

• Keith Conrad, The Congruent Number Problem, The Harvard College Mathematics Review, 2008.

Crossrefs

Cf. A222946, A225949, A249866, A249869.

Formula

T(n, m) = 2*n*m - (n^2 - m^2) if n > m >= 1 and gcd(n, m) = 1, n+m odd, and T(n, m) = 0 otherwise.