A225951 -id:A225951 - cp's OEIS frontend

A222946 Triangle for hypotenuses of primitive Pythagorean triangles.

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.

A249869 Triangle giving the area of primitive Pythagorean triangles, with zero entries for non-primitive triangles.

Original entry on oeis.org

6, 0, 30, 60, 0, 84, 0, 210, 0, 180, 210, 0, 0, 0, 330, 0, 630, 0, 924, 0, 546, 504, 0, 1320, 0, 1560, 0, 840, 0, 1386, 0, 2340, 0, 0, 0, 1224, 990, 0, 2730, 0, 0, 0, 3570, 0, 1710, 0, 2574, 0, 4620, 0, 5610, 0, 5016, 0, 2310, 1716, 0, 0, 0, 7140, 0, 7980, 0, 0, 0, 3036

Offset: 2

Wolfdieter Lang, Dec 03 2014

See A249866 for comments and references.
For the sorted areas of all primitive Pythagorean triangles (x, y, z) with, say y even, see A024406.
Note that in a row > N there may appear smaller numbers than the maximal number up to row N. Therefore the sorted nonvanishing numbers up to a given row N will in general not produce a subsequence of A024406. The minimal areas in rows n = 2..20 are 6, 30, 60, 180, 210, 546, 504, 1224, 990, 2310, 1716, 3900, 2730, 6090, 4080, 8976, 5814, 12654, 7980. For example, one has to go up to row n = 16 to cover all areas <= 4080.
See the link for more details on a safe row number n = N to cover all areas not exceeding a given one, and also for all areas <= 10^6 with their squarefree parts. - Wolfdieter Lang, Nov 25 2016

			The triangle T(n, m) begins:
n\m    1    2    3     4     5     6    7     8     9   10    11
2:     6
3:     0   30
4:    60    0   84
5:     0  210    0   180
6:   210    0    0     0   330
7:     0  630    0   924     0   546
8:   504    0 1320     0  1560     0  840
9:     0 1386    0  2340     0     0    0 1224
10:  990    0 2730     0     0     0 3570    0   1710
11:    0 2574    0  4620     0  5610    0 5016      0 2310
12: 1716    0    0     0  7140     0 7980     0     0    0  3036
...
For more rows see the link.
T(5, 2) = 210 for the primitive triangle (21, 20, 29).
T(6, 1) = 210 for the primitive triangle (35, 12, 37).

Wolfdieter Lang, First rows of the triangle.
Wolfdieter Lang, A Note on the Area table A249869 for Primitive Pythagorean Triangles.

Cf. A024406, A249866, A258150 (one sixth of this triangle), A225949 (leg sums), A225951 (perimeters), A222946 (hypotenuses), A208854 (odd catheti), A208855 (even catheti), A278711.

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

A225950 Triangle for odd legs of primitive Pythagorean triangles.

Original entry on oeis.org

3, 0, 5, 15, 0, 7, 0, 21, 0, 9, 35, 0, 0, 0, 11, 0, 45, 0, 33, 0, 13, 63, 0, 55, 0, 39, 0, 15, 0, 77, 0, 65, 0, 0, 0, 17, 99, 0, 91, 0, 0, 0, 51, 0, 19, 0, 117, 0, 105, 0, 85, 0, 57, 0, 21, 143, 0, 0, 0, 119, 0, 95, 0, 0, 0, 23, 0, 165, 0, 153, 0, 133, 0, 105, 0, 69, 0, 25, 195, 0, 187, 0, 171, 0, 0, 0, 115, 0, 75, 0, 27, 0, 221, 0, 209, 0, 0, 0, 161, 0, 0, 0, 0, 0, 29

Offset: 2

Wolfdieter Lang, May 23 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 (for these solutions).
The number of non-vanishing entries in row n is A055034(n).
The sequence of the main diagonal is 2*n -1 = A005408(n-1),
   n >= 2.
If the zeros are eliminated and the numbers are sorted nondecreasingly (multiple entries appear) one obtains A120890. All odd numbers >= 3 appear, they are given in A005408. Note that all odd legs x will be found if one takes in the triangle n = 2, ..., floor((x+1)/2).

			The triangle a(n,m) begins:
n\m   1    2   3    4    5    6   7     8   9   10  11  12 ...
2:    3
3:    0    5
4:   15    0   7
5:    0   21   0    9
6:   35    0   0    0   11
7:    0   45   0   33    0   13
8:   63    0  55    0   39    0  15
9:    0   77   0   65    0    0   0    17
10:  99    0  91    0    0    0  51     0  19
11:   0  117   0  105    0   85   0    57   0   21
12: 143    0   0    0  119    0  95     0   0    0  23
13:   0  165   0  153    0  133   0   105   0   69   0  25
...
a(6,1) = 35 from the primitive triangle (35,12,37).
a(6,2) = 0 because n and m are even (not allowed n, m values for primitive triangles).
a(6,3) = 0 because gcd(6,3) = 3 (not 1, hence not allowed).

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.

Kival Ngaokrajang, Illustration of pattern of zero terms (non-isolated zeros are colored), for n = 1..50.

Cf. A222946 (hypotenuses), A225952 (even legs), A225949 (leg sums), A225951 (perimeters), A120890 (odd legs, ordered).

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.

A225952 Triangle read by rows, giving the even legs of primitive Pythagorean triangles, with zero entries for non-primitive triangles.

Original entry on oeis.org

4, 0, 12, 8, 0, 24, 0, 20, 0, 40, 12, 0, 0, 0, 60, 0, 28, 0, 56, 0, 84, 16, 0, 48, 0, 80, 0, 112, 0, 36, 0, 72, 0, 0, 0, 144, 20, 0, 60, 0, 0, 0, 140, 0, 180, 0, 44, 0, 88, 0, 132, 0, 176, 0, 220, 24, 0, 0, 0, 120, 0, 168, 0, 0, 0, 264, 0, 52, 0, 104, 0, 156, 0, 208, 0, 260, 0, 312, 28, 0, 84, 0, 140, 0, 0, 0, 252, 0, 308, 0, 364

Offset: 2

Wolfdieter Lang, May 23 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.
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) = 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*(n-1) = 4*A000217 (n-1), n >= 2.
If the 0 entries are eliminated and the numbers are ordered nondecreasingly (multiple entries appear) the sequence becomes A120427. All its entries are positive integer multiples of 4, shown in  A008586(n), n >= 1. Note that all even legs <= N are certainly reached if one considers in the triangle rows n = 2, ..., floor(N/2).

			The triangle a(n,m) begins:
n\m   1   2   3   4    5    6    7    8    9    10   11 ...
2:    4
3:    0  12
4:    8   0  24
5:    0  20   0  40
6:   12   0   0   0   60
7:    0  28   0  56    0   84
8:   16   0  48   0   80    0  112
9:    0  36   0  72    0    0    0  144
10:  20   0  60   0    0    0  140    0  180
11:   0  44   0  88    0  132    0  176    0   220
12:  24   0   0   0  120    0  168    0    0     0  264
...

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.

Kival Ngaokrajang, Illustration of pattern of zero terms (non-isolated zeros are colored), for n = 1..103.

Cf. A222946 (hypotenuses), A225950 (odd legs), A225949 (leg sums), A225951 (perimeters), A120427 (even legs ordered), A008586 (multiples of 4).

a(n,m) = 2*n*m 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.

Edited. Refs. added. - Wolfdieter Lang, Jul 26 2014

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A222946 Triangle for hypotenuses of primitive Pythagorean triangles.

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A249869 Triangle giving the area of primitive Pythagorean triangles, with zero entries for non-primitive triangles.

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A225950 Triangle for odd legs of primitive Pythagorean triangles.

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A225952 Triangle read by rows, giving the even legs of primitive Pythagorean triangles, with zero entries for non-primitive triangles.

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