A258150 -id:A258150 - cp's OEIS frontend

A024406 Ordered areas of primitive Pythagorean triangles.

6, 30, 60, 84, 180, 210, 210, 330, 504, 546, 630, 840, 924, 990, 1224, 1320, 1386, 1560, 1710, 1716, 2310, 2340, 2574, 2730, 2730, 3036, 3570, 3900, 4080, 4290, 4620, 4914, 5016, 5610, 5814, 6090, 6630, 7140, 7440, 7854, 7956, 7980, 7980, 8970, 8976, 9690

Offset: 1

David W. Wilson

This sequence also gives Fibonacci's congruous numbers (or congrua) divided by 4 with multiplicities, not regarding leg exchange in the underlying primitive Pythagorean triangle. See A258150 and the example. - Wolfdieter Lang, Jun 14 2015

The squarefree part of an entry which is not squarefree is a primitive congruent number from A006991 belonging to a Pythagorean triangle with rational (not all integer) side lengths (and its companion obtained by exchanging the legs). See the W. Lang link. - Wolfdieter Lang, Oct 25 2016

			a(6) = a(7) = 210 corresponds to the area (in some squared length unit) of the primitive Pythagorean triangles (21, 20, 29) and (35, 12, 37). Fibonacci's congruum C = 840 = 210*4 belongs to the two triples [x, y, z] = [29, 41, 1] and [37, 47, 23], solving x^2 + C = y^2 and x^2 - C = z^2. - _Wolfdieter Lang_, Jun 14 2015
a(5) = 180 = 6^2*5 lead to the primitive congruent number A006991(1) = 5 from the primitive Pythagorean triangle [9, 40, 41] after division by 6: [3/2, 20/3, 41/6]. See the link for the other nonsquarefree a(n) numbers. - _Wolfdieter Lang_, Oct 25 2016

Giovanni Resta, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triples and Online Calculators
Wolfdieter Lang, Non-squarefree entries, their congruent numbers and rational Pythagorean triangles
Eric Weisstein's World of Mathematics, Congruum Problem

Cf. A009112, A024365, A094182, A094183, A256418 (congrua), A258150.

a(n) = 6*A020885(n). - Lekraj Beedassy, Apr 30 2004

a(n) = A121728(n)*A121729(n)/2. - M. F. Hasler, Apr 16 2020

A024365 Areas of right triangles with coprime integer sides.

Original entry on oeis.org

6, 30, 60, 84, 180, 210, 330, 504, 546, 630, 840, 924, 990, 1224, 1320, 1386, 1560, 1710, 1716, 2310, 2340, 2574, 2730, 3036, 3570, 3900, 4080, 4290, 4620, 4914, 5016, 5610, 5814, 6090, 6630, 7140, 7440, 7854, 7956, 7980, 8970, 8976, 9690, 10374

Offset: 1

David W. Wilson

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives areas A*B/2.
By Theorem 2 of Mohanty and Mohanty, all these numbers are primitive Pythagorean. - T. D. Noe, Sep 24 2013
This sequence also gives Fibonacci's congruous numbers (without multiplicity, in increasing order) divided by 4. See A258150. - Wolfdieter Lang, Jun 14 2015
The same as A024406 with duplicates removed. All terms are multiples of 6, cf. A258151. - M. F. Hasler, Jan 20 2019

			6 is in the sequence because it is the area of the 3-4-5 triangle.
a(7) = 210 corresponds to the two primitive Pythagorean triangles (21, 20, 29) and (35, 12, 37). See A024406. - _Wolfdieter Lang_, Jun 14 2015

T. D. Noe, Table of n, a(n) for n = 1..10000 (corrected by _Giovanni Resta_, Jan 21 2019)
Supriya Mohanty and S. P. Mohanty, Pythagorean Numbers, Fibonacci Quarterly 28 (1990), 31-42.

Cf. A009111, A009112, A024406 (with multiplicity), A258150, A024407, A258151 (terms divided by 6).

Subsequence of A073120 and A147778.

nn = 22; (* nn must be even *) t = Union[Flatten[Table[If[GCD[u, v] == 1 && Mod[u, 2] + Mod[v, 2] == 1, u v (u^2 - v^2), 0], {u, nn}, {v, u - 1}]]]; Select[Rest[t], # < nn (nn^2 - 1) &] (* T. D. Noe, Sep 19 2013 *)

select( {is_A024365(n)=my(N=1+#n=divisors(2*n)); for(i=1, N\2, gcd(n[i], n[N-i])==1 && issquare(n[i]^2+n[N-i]^2) && return(n[i]))}, [1..10^4]) \\ is_A024365 returns the smaller leg if n is a term, else 0. - M. F. Hasler, Jun 06 2024

Positive integers of the form u*v*(u^2 - v^2) where 2uv and u^2 - v^2 are coprime or, alternatively, where u, v are coprime and one of them is even.

a(n) = 6*A258151(n). - M. F. Hasler, Jan 20 2019

Additional comments James R. Buddenhagen, Aug 10 2008 and from Max Alekseyev, Nov 12 2008

Edited by N. J. A. Sloane, Nov 20 2008 at the suggestion of R. J. Mathar

A020885 Ordered areas (divided by 6) of primitive Pythagorean triangles (with multiple entries).

Original entry on oeis.org

1, 5, 10, 14, 30, 35, 35, 55, 84, 91, 105, 140, 154, 165, 204, 220, 231, 260, 285, 286, 385, 390, 429, 455, 455, 506, 595, 650, 680, 715, 770, 819, 836, 935, 969, 1015, 1105, 1190, 1240, 1309, 1326, 1330, 1330, 1495, 1496, 1615, 1729, 1771, 1785, 1820, 1925

Offset: 1

Clark Kimberling

nonn

Since squares are 0 or 1 under both mod 3 and mod 4, for the Pythagorean equation A^2 + B^2 = C^2 to hold, each of 3 and 4 divides either of leg A or leg B, so that area A*B/2 is divisible by 3*4/2 = 6. - Lekraj Beedassy, Apr 30 2004
From Wolfdieter Lang, Jun 14 2015: (Start)
This sequence gives the area/6 (in some squared length unit) of primitive Pythagorean triangles with multiplicities modulo leg exchange. See the example.
This sequence also gives Fibonacci's congruous numbers divided by 24, with multiplicities and ordered nondecreasingly. See A258150.
(End)
It appears that this sequence gives the list of dimensions of irreducible unitary representations of the Lie group SO(5). - Antoine Bourget, Mar 30 2022

			a(6) = a(7) = 35 from the two Pythagorean triangles (A,B,C) = (21, 20, 29)  and (35, 12, 37) with area 210. Triangles (20, 21, 29) and (12, 35, 37) are not counted (leg exchange). - _Wolfdieter Lang_, Jun 14 2015

Giovanni Resta, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triples and Online Calculators

Cf. A020882, A020883, A020884, A020886.

Take[Sort[(Times@@#)/12&/@({Times@@#,(Last[#]^2-First[#]^2)/2}&/@ Select[ Subsets[Range[1,41,2],{2}],GCD@@#==1&])],60] (* Harvey P. Dale, Feb 27 2012 *)

a(n) = A024406(n)/6.

Extended and corrected by David W. Wilson

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.

A258149 Triangle of the absolute difference of the two legs (catheti) of primitive Pythagorean triangles.

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

Offset: 2

Wolfdieter Lang, Jun 10 2015

For primitive Pythagorean triangles characterized by certain (n,m) pairs and references see A225949.
Here a(n,m) = 0 for non-primitive Pythagorean triangles, and for primitive Pythagorean triangles a(n,m) = abs(n^2 - m^2 - 2*n*m) = abs((n-m)^2 - 2*m^2).
The number of non-vanishing entries in row n is A055034(n).
D(n,m):= n^2 - m^2 - 2*n*m >= 0 if 1 <= m <= floor(n/(sqrt(2)+1)), and D(n,m) < 0 if n/(sqrt(2)+1)+1 <= m <= n-1, for n >= 2.
The Pell equation (n-m)^2 - 2*m^2 = +/- N is important here to find the representations of +N or -N in the triangle D(n,m). For instance, odd primes N have to be of the +1 (mod 8) (A007519) or -1 (mod 8) (A007522) form, that is, from A001132. See the Nagell reference, Theorem 110, p. 208 with Theorem 111, pp. 210-211. E.g., N = +7 appears for m = 1, 3, 9, 19, 53, ... (A077442) for n = 4, 8, 22, 46, 128, ... (2*A006452).
  N = -7 appears for n = 3, 9, 19, 53, 111, ... (A077442) and m = 2, 4, 8, 22, 46, ... (2*A006452).
For the  signed version 2*n*m - (n^2 - m^2) see A278717. - 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:    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
12: 119  0  0  0  1  0 73   0   0   0 241
...
a(2,1) = |1^2 - 2*1^2| = 1 for the primitive Pythagorean triangle (pPt) [3,4,5] with |3-4| = 1.
a(3,2) = |1^2 - 2*2^2| = 7 for the pPt [5,12,13] with |5 - 12| = 7.
a(4,1) = |3^2 - 2*1^2| = 7 for the pPt [15, 8, 17] with |15 - 8| = 7.

See also A225949.
T. Nagell, Introduction to Number Theory, Chelsea Publishing Company, New York, 1964, pp. 208, 210-211.

Cf. A249866, A222946, A225949, A222951, A258150, A278717 (signed).

a[n_, m_] /; n > m >= 1 && CoprimeQ[n, m] && (-1)^(n+m) == -1 := Abs[n^2 - m^2 - 2*n*m]; a[, ] = 0; Table[a[n, m], {n, 2, 12}, {m, 1, n-1}] // Flatten (* Jean-François Alcover, Jun 16 2015, after given formula *)

a(n,m) = abs(n^2 - m^2 -2*n*m) = abs((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.

A258151 Areas of primitive Pythagorean triangles divided by 6, in increasing order without multiple entries.

Original entry on oeis.org

1, 5, 10, 14, 30, 35, 55, 84, 91, 105, 140, 154, 165, 204, 220, 231, 260, 285, 286, 385, 390, 429, 455, 506, 595, 650, 680, 715, 770, 819, 836, 935, 969, 1015, 1105, 1190, 1240, 1309, 1326, 1330, 1495, 1496, 1615, 1729, 1771, 1785, 1820, 1925

Offset: 1

Wolfdieter Lang, Jun 14 2015

See A020885 for this sequence with multiplicities. See A024365 for the areas with multiplicities.

This sequence gives also Fibonacci's congruous numbers divided by 24 without multiple entries. See A258150.

			See A020885.

Cf. A020885, A024365, A258150.

cp's OEIS Frontend

A024406 Ordered areas of primitive Pythagorean triangles.

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A024365 Areas of right triangles with coprime integer sides.

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A020885 Ordered areas (divided by 6) of primitive Pythagorean triangles (with multiple entries).

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

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A258149 Triangle of the absolute difference of the two legs (catheti) of primitive Pythagorean triangles.

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A258151 Areas of primitive Pythagorean triangles divided by 6, in increasing order without multiple entries.

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