A249869 -id:A249869 - cp's OEIS frontend

A278711 Triangle T read by rows: T(n, m), for n >= 2, and m=1, 2, ..., n-1, equals the positive integer solution x of y^2 = x^3 - A(n, m)^2*x with the area A(n, m) = A249869(n, m) of the primitive Pythagorean triangle characterized by (n, m) or 0 if no such triangle exists.

12, 0, 45, 240, 0, 112, 0, 525, 0, 225, 1260, 0, 0, 0, 396, 0, 2205, 0, 1617, 0, 637, 4032, 0, 3520, 0, 2496, 0, 960, 0, 6237, 0, 5265, 0, 0, 0, 1377, 9900, 0, 9100, 0, 0, 0, 5100, 0, 1900, 0, 14157, 0, 12705, 0, 10285, 0, 6897, 0, 2541, 20592, 0, 0, 0, 17136, 0, 13680, 0, 0, 0, 3312, 0, 27885, 0, 25857, 0, 22477, 0, 17745, 0, 11661, 0, 4225, 38220, 0, 36652, 0, 33516, 0, 0, 0, 22540, 0, 14700, 0, 5292, 0, 49725, 0, 47025, 0, 0, 0, 36225, 0, 0, 0, 0, 0, 6525

Offset: 2

Wolfdieter Lang, Nov 27 2016

The corresponding triangle with the square root of the positive integer solutions y is A278712.
A primitive Pythagorean triangle is characterized by two integers n > m >= 1, gcd(n, m) = 1 and n+m odd. See A249866, also for references.
For the one-to-one correspondence between rational Pythagorean triangles with area A > 0 and rational points on the elliptic curve y^2 = x^3 - A^2*x with y not vanishing see Theorem 4.1 of the Keith Conrad link or Theorem 15.6, p. 212, of the Ash-Gross reference.

			The triangle T(n, m) begins:
  n\m     1    2    3    4    5   6    7    8
  2:     12
  3:      0   45
  4:    240    0  112
  5:      0  525    0  225
  6:   1260    0    0    0  396
  7:      0 2205    0 1617    0 637
  8:   4032    0 3520    0 2496   0  960
  9       0 6237    0 5265    0   0    0 1377
  ...........................................
  n = 10: 9900 0 9100 0 0 0 5100 0 1900,
  n = 11: 0 14157 0 12705 0 10285 0 6897 0 2541,
  n = 12: 20592 0 0 0 17136 0 13680 0 0 0 3312,
  n = 13: 0 27885 0 25857 0 22477 0 17745 0 11661 0 4225,
  n = 14: 38220 0 36652 0 33516 0 0 0 22540 0 14700 0 5292,
  n = 15: 0 49725 0 47025 0 0 0 36225 0 0 0 0 0 6525.
  ...
The triangle of solutions [x,y] begins ([0,0] if there is no primitive Pythagorean):
  n\m        1           2         3          4
  2:   [12,36]
  3:     [0,0]    [45,225]
  4:[240,3600]       [0,0] [112,784]
  5:     [0,0] [525,11025]     [0,0] [225, 2025]
  ...
  n=6: [1260,44100] [0,0] [0,0] [0,0] [396,4356],
  n=7: [0,0] [2205,99225] [0,0] [1617,53361] [0.0] [637,8281],
  n=8: [4032,254016] [0,0] [3520,193600] [0,0] [2496,97344] [0,0] [960,14400],
  n=9: [0,0] [6237,480249] [0,0] [5265,342225] [0,0] [0,0] [0,0] [1377,23409],
  n=10: [9900,980100] [0,0] [9100,828100] [0,0] [0,0] [0,0] [5100,260100] [0,0] [1900, 36100].
  ...

Avner Ash and Robert Gross, Elliptic tales: curves, counting, and number theory, Princeton University Press, 2012.
Keith Conrad, The Congruent Number Problem, The Harvard College Mathematics Review, 2008.

Cf. A249866, A249869, A278712.

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

A278712 Triangle T read by rows: T(n, m), for n >= 2, and m = 1, 2, ..., n-1, equals the square root of the positive integer solution y of y^2 = x^3 - A(n, m)^2*x with the area A(n, m) = A249869(n, m) of the primitive Pythagorean triangle characterized by (n, m) or 0 if no such triangle exists.

Original entry on oeis.org

6, 0, 15, 60, 0, 28, 0, 105, 0, 45, 210, 0, 0, 0, 66, 0, 315, 0, 231, 0, 91, 504, 0, 440, 0, 312, 0, 120, 0, 693, 0, 585, 0, 0, 0, 153, 990, 0, 910, 0, 0, 0, 510, 0, 190, 0, 1287, 0, 1155, 0, 935, 0, 627, 0, 231, 1716, 0, 0, 0, 1428, 0, 1140, 0, 0, 0, 276, 0, 2145, 0, 1989, 0, 1729, 0, 1365, 0, 897, 0, 325, 2730, 0, 2618, 0, 2394, 0, 0, 0, 1610, 0, 1050, 0, 378, 0, 3315, 0, 3135, 0, 0, 0, 2415, 0, 0, 0, 0, 0, 435

Offset: 2

Wolfdieter Lang, Nov 27 2016

The corresponding solutions x are given in A278711, where also details are found.

			The triangle T(n, m) begins:
n\m   1    2   3    4   5   6   7   8   9  10
2:    6
3:    0   15
4:   60    0  28
5:    0  105   0   45
6:  210    0   0    0  66
7:    0  315   0  231   0  91
8:  504    0 440    0 312   0 120
9:    0  693   0  585   0   0   0 153
10: 990    0 910    0   0   0 510   0 190
11:   0 1287   0 1155   0 935   0 627   0 231
...
n = 12: 1716 0 0 0 1428 0 1140 0 0 0 276,
n = 13: 0 2145 0 1989 0 1729 0 1365 0 897 0 325,
n = 14: 2730 0 2618 0 2394 0 0 0 1610 0 1050 0 378,
n = 15: 0 3315 0 3135 0 0 0 2415 0 0 0 0 0 435.
...
For the solutions [x,y] see A278711.

Cf. A278711.

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

A222946 Triangle for hypotenuses of primitive Pythagorean triangles.

Original entry on oeis.org

5, 0, 13, 17, 0, 25, 0, 29, 0, 41, 37, 0, 0, 0, 61, 0, 53, 0, 65, 0, 85, 65, 0, 73, 0, 89, 0, 113, 0, 85, 0, 97, 0, 0, 0, 145, 101, 0, 109, 0, 0, 0, 149, 0, 181, 0, 125, 0, 137, 0, 157, 0, 185, 0, 221, 145, 0, 0, 0, 169, 0, 193, 0, 0, 0, 265, 0, 173, 0, 185, 0, 205, 0, 233, 0, 269, 0, 313, 197, 0, 205, 0, 221, 0, 0, 0, 277, 0, 317, 0, 365

Offset: 2

Wolfdieter Lang, Mar 21 2013

For primitive Pythagorean triples (x,y,z) see the Niven et al. reference, Theorem 5.5, p. 232, and the Hardy-Wright reference, Theorem 225, p. 190.
Here a(n,m) = 0 for non-primitive Pythagorean triangles.
There is a one-to-one correspondence between the values n and m of this number triangle for which a(n,m) does not vanish and primitive solutions of x^2 + y^2 = z^2 with y even, namely x = n^2 - m^2, y = 2*n*m and z = n^2 + m^2.
The diagonal sequence is given by a(n,n-1) = A001844(n-1), n >= 2.
The row sums of this triangle are 5, 13, 42, 70, 98, 203, 340, 327, 540, ...
a(n,k) = A055096(n-1,k) * ((n+k) mod 2) * A063524 (gcd(n,k)): terms in A055096 that are not hypotenuses in primitive Pythagorean triangles, are replaced by 0. - Reinhard Zumkeller, Mar 23 2013
The number of non-vanishing entries in row n is A055034(n). - Wolfdieter Lang, Mar 24 2013
The non-vanishing entries when ordered according to nondecreasing leg sums x+y (see A225949 and A198441) produce (with multiplicities) A198440. - Wolfdieter Lang, May 22 2013
a(n, m) also gives twice the member s(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, or 0 if there is no such triangle. The other members are given by 2*r(n, m) = A278717(n, m) and 2*t(n, m) = A225949(n, m). See A278717 for details and the Keith Conrad reference there. - Wolfdieter Lang, Nov 30 2016

			The triangle a(n,m) begins:
n\m   1   2   3   4   5   6   7   8   9  10  11  12   13 ...
2:    5
3:    0  13
4:   17   0  25
5:    0  29   0  41
6:   37   0   0   0  61
7:    0  53   0  65   0  85
8:   65   0  73   0  89   0 113
9:    0  85   0  97   0   0   0 145
10: 101   0 109   0   0   0 149   0 181
11:   0 125   0 137   0 157   0 185   0 221
12: 145   0   0   0 169   0 193   0   0   0 265
13:   0 173   0 185   0 205   0 233   0 269   0 313
14: 197   0 205   0 221   0   0   0 277   0 317   0  365
...
------------------------------------------------------------
a(7,4) = 7^2 + 4^2 = 49 + 16 = 65.
a(8,1) = 8^2 + 1^2 = 64 +  1 = 65.
a(3,1) = 0 because n and m are both odd.
a(4,2) = 0 because n and m are both even.
a(6,3) = 0 because gcd(6,3) = 3 (not 1).
The primitive triangle for (n,m) = (2,1) is (x,y,z) = (3,4,5).
The primitive triangle for (n,m) = (7,4) is (x,y,z) = (33,56,65).
The primitive triangle for (n,m) = (8,1) is (x,y,z) = (63,16,65).

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003.
Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991.

Reinhard Zumkeller, Rows n = 2..120 of triangle, flattened

Cf. A020882 (ordered nonzero values a(n,m) with multiplicity).

Cf. A249866, A225950 (odd legs), A225951 (perimeters), A225952 (even legs), A225949 (leg sums), A249869 (areas), A258149 (absolute leg differences), A278717 (leg differences).

a222946 n k = a222946_tabl !! (n-2) !! (k-1)
a222946_row n = a222946_tabl !! (n-2)
a222946_tabl = zipWith p [2..] a055096_tabl where
   p x row = zipWith (*) row $
             map (\k -> ((x + k) `mod` 2) * a063524 (gcd x k)) [1..]
-- Reinhard Zumkeller, Mar 23 2013

a(n,m) = n^2 + m^2 if n > m >= 1, gcd(n,m) = 1, and n and m are integers of opposite parity (i.e., (-1)^(n+m) = -1), otherwise a(n,m) = 0.

A225949 Triangle for sum of the two legs (catheti) of primitive Pythagorean triangles.

Original entry on oeis.org

7, 0, 17, 23, 0, 31, 0, 41, 0, 49, 47, 0, 0, 0, 71, 0, 73, 0, 89, 0, 97, 79, 0, 103, 0, 119, 0, 127, 0, 113, 0, 137, 0, 0, 0, 161, 119, 0, 151, 0, 0, 0, 191, 0, 199, 0, 161, 0, 193, 0, 217, 0, 233, 0, 241, 167, 0, 0, 0, 239, 0, 263, 0, 0, 0, 287, 0, 217, 0, 257, 0, 289, 0, 313, 0, 329, 0, 337, 223, 0, 271, 0, 311, 0, 0, 0, 367, 0, 383, 0, 391

Offset: 2

Wolfdieter Lang, May 21 2013

For primitive Pythagorean triples (x,y,z) see the Niven et al. reference, Theorem 5.5, p. 232, and the Hardy-Wright reference, Theorem 225, p. 190.
Here a(n,m) = 0 for non-primitive Pythagorean triangles.
There is a one-to-one correspondence between the values n and m of this number triangle for which a(n,m) does not vanish and primitive solutions of x^2 + y^2 = z^2 with y even, namely x = n^2 - m^2, y = 2*n*m and z = n^2 + m^2. The mirror triangles with x even are not considered here. Therefore a(n,m) = n^2 - m^2 + 2*n*m (for these solutions).
The number of non-vanishing entries in row n is A055034(n).
The sequence of the main diagonal is 2*n^2-1 = A056220(n), n>= 2.
The sequence of the main diagonal is j^2 + k^2 - 2 or 2*j*k if n>=2 and j = n + sqrt(2)/2 and k = n - sqrt(2)/2. - Avi Friedlich, Mar 30 2015
If the 0 entries are eliminated and the numbers are ordered increasingly (keeping multiple entries) the sequence becomes A198441(n-1), n>=2. If multiple entries are recorded only once this becomes A058529 (a proper subsequence of A118905). Note that all leg sums <= N are certainly reached if one considers rows n = 2, ..., floor(-1 + sqrt(N+2)).
a(n, m) also gives twice the member t(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, or 0 if there is no such triangle. The other members are given by 2*r(n, m) = A278717(n, m) and 2*s(n, m) = A222946(n, m). See A278717 for details and the Keith Conrad reference. - Wolfdieter Lang, Nov 30 2016

			The triangle a(n,m) begins:
n\m   1   2   3   4   5   6    7    8    9   10   11 ...
2:    7
3:    0  17
4:   23   0  31
5:    0  41   0  49
6:   47   0   0   0  71
7:    0  73   0  89   0  97
8:   79   0 103   0 119   0  127
9:    0 113   0 137   0   0    0  161
10: 119   0 151   0   0   0  191    0  199
11:   0 161   0 193   0 217    0  233    0  241
12: 167   0   0   0 239   0  263    0    0    0  287
...
---------------------------------------------------------
The primitive triangle for (n,m) = (2,1) is (x,y,z) = (3,4,5), with a(2,1) = 3 + 4 = 7.
The primitive triangle for (n,m) = (7,4) is (x,y,z) = (33,56,65), with a(7,4) = 33 + 56 = 89.
The primitive triangle for (n,m) = (8,1) is (x,y,z) = (63,16,65), with a(8,1) = 63 + 16 = 79.
All primitive Pythagorean triangles with leg sums <= 167 are certainly covered by this triangle (rows n = 2..12), and the multiplicities are also correct, e.g., 119 appears twice.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth Edition, Clarendon Press, Oxford, 2003.
Ivan Niven, Herbert S. Zuckerman and Hugh L. Montgomery, An Introduction to the Theory Of Numbers, Fifth Edition, John Wiley and Sons, Inc., NY 1991.

Cf. A222946 (hypotenuses), A222951 (perimeters), A056220 (main diagonals), A198441 (no zeros, ordered), A258149 (absolute leg differences), A278717 (leg differences).

T[n_, m_] := If[n > m >= 1 && GCD[n, m] == 1 && (-1)^(n+m) == -1, (n+m)^2 - 2 m^2, 0];
Table[T[n, m], {n, 2, 14}, {m, 1, n-1}] // Flatten (* Jean-François Alcover, Oct 22 2021 *)

a(n,m) = (n+m)^2 - 2*m^2 if n > m >= 1, gcd(n,m) = 1, and n and m are integers of opposite parity (i.e., (-1)^(n+m) = -1); otherwise a(n,m) = 0.

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).

Original entry on oeis.org

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

Wolfdieter Lang, Nov 29 2016

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.

			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).
...

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

Cf. A222946, A225949, A249866, A249869.

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.

A278147 Triangle read by rows of Cantor pairing function value determining primitive Pythagorean triangles or 0 if there is no such triangle.

Original entry on oeis.org

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

Offset: 2

Wolfdieter Lang, Nov 21 2016

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.

			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.
...

Cf. A277557, A249869.

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.

cp's OEIS Frontend

A278711 Triangle T read by rows: T(n, m), for n >= 2, and m=1, 2, ..., n-1, equals the positive integer solution x of y^2 = x^3 - A(n, m)^2*x with the area A(n, m) = A249869(n, m) of the primitive Pythagorean triangle characterized by (n, m) or 0 if no such triangle exists.

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A278712 Triangle T read by rows: T(n, m), for n >= 2, and m = 1, 2, ..., n-1, equals the square root of the positive integer solution y of y^2 = x^3 - A(n, m)^2*x with the area A(n, m) = A249869(n, m) of the primitive Pythagorean triangle characterized by (n, m) or 0 if no such triangle exists.

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A222946 Triangle for hypotenuses of primitive Pythagorean triangles.

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Haskell

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A225949 Triangle for sum of the two legs (catheti) of primitive Pythagorean triangles.

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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).

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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.

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