A010814 -id:A010814 - cp's OEIS frontend

A156687 Perimeters of Pythagorean triangles that can be constructed in exactly 5 different ways.

420, 660, 924, 1008, 1080, 1200, 1512, 1584, 1716, 1800, 1872, 1890, 2700, 3150, 3168, 3240, 3480, 3528, 3570, 3720, 3744, 4410, 4440, 4536, 4590, 4704, 4872, 4896, 4950, 5208, 5292, 5472, 5600, 5670, 6000, 6090, 6210, 6216, 6624, 6630, 6660, 6888

Offset: 1

Ant King, Feb 18 2009

For any given N we can always find at least N Pythagorean triangles with the same perimeter.

			As 924 is the third smallest integer that can occur as the perimeter of exactly 5 Pythagorean triples - specifically (42,440,442), (77,420,427), (132,385,407), (198,336,390) and (231,308,385) - then a(3)=924.

Sierpinski, W.; Pythagorean Triangles, Dover Publications, Inc., Mineola, New York, 2003.
Beiler, Albert H.; Recreations In The Theory Of Numbers, Chapter XIV, The Eternal Triangle, Dover Publications Inc., New York, 1964, pp. 104-134.

Ray Chandler, Table of n, a(n) for n = 1..10000
Ron Knott, Right-angled Triangles and Pythagoras' Theorem

Cf. A098714, A099831, A099832, A099833, A009129, A010814.

SetSystemOptions["ReduceOptions"->{"DiscreteSolutionBound"->100000}];AllPerimeterTriples[n_Integer]/;n>0:=Module[{result=Reduce[Reduce[{x^2+y^2==z^2,z>y>x>0,Element[{x,y,z},Integers],x+y+z==n},{x,y,z}]]},If[result===False,{},Sort[{x,y,z}/.{ToRules[result]}]]];Select[Range[10000],Length[AllPerimeterTriples[ # ]]==5 &]

A382268 Numbers k such that a right triangle can be formed from a chain of linked rods of lengths 1, 2, 3, ..., k, with the perimeter equal to the total length.

Original entry on oeis.org

15, 20, 24, 35, 39, 44, 48, 55, 56, 63, 75, 76, 80, 84, 91, 95, 99, 104, 111, 119, 120, 132, 135, 140, 143, 144, 152, 155, 168, 175, 176, 187, 188, 195, 203, 207, 215, 216, 219, 224, 252, 259, 260, 264, 272, 275, 279, 287, 288, 296, 299, 308, 315, 320, 324, 335, 351, 360

Offset: 1

Ali Sada and Daniel Mondot, Mar 19 2025

nonn

The corresponding perimeters T(a(n)) must be in the intersection of T = A000217 (triangular numbers) with A010814 (perimeters of integer sided right triangles). A number k is in the sequence if there exists a solution {k1, k2, k3} with k > k1 > k2 > k3 >= 0 such that for a < b < c in { T(k1) - T(k2), T(k2) - T(k3), T(k) - T(k1) + T(k3) } one has a^2 + b^2 = c^2. - M. F. Hasler, Mar 20 2025

			The first triangle is when k = 15. The segments are [6+7+8+9+10] [11+12+13+14] [15+1+2+3+4+5]. The sums of the segments are (40, 50, 30), which is 10 times the primitive right triangle (3, 4, 5).
The second term, k = 20, corresponds to 5 distinct solutions:
  S1 = {18, 16, 9}: a = 9+...+1 + 20+19 = 84, b = 18+17 = 35, c = 16+...+10 = 91,
  S2 = {17, 11, 3}: a = 20+19+18 + 3+2+1 = 63, c = 17+...+12 = 87, b = 11+...+4 = 60,
  S3 = {17, 11, 2}: a = 20+19+18 + 2+1 = 60, c = 17+...+12 = 87, b = 11+...+3 = 63,
  S4 = {16, 9, 4}: a = 20+...+17 + 4+...+1 = 84, c = 16+...+10 = 91, b = 9+...+5 = 35,
  S5 = {15, 8, 1}: c = 20+...+16 + 1 = 91, a = 15+...+9 = 84, b = 8+...+2 = 35.
We note that S2 and S3, and S1, S4 and S5, have the same side lengths, but different decompositions.

Daniel Mondot, Table of n, a(n) for n = 1..3052

Cf. A000217, A010814, A380867, A380868, A380875.

select( {is_A382268(n)=my(Tn=n*(n+1)\2,T1,T2,S); Tn%2==0 && is_A005279(Tn\2) && forstep(n1=n-1,sqrtint(Tn-1)+1,-1, T1=n1*(n1+1)\2; forstep(n2=n1-1,sqrtint(2*T1-Tn-1)+1,-1, T2=n2*(n2+1)\2; forstep(n3=n2-1,0,-1, #(S=Set([Tn-T1+S=n3*(n3+1)\2,T2-S,T1-T2]))>2 && S[3]^2 == S[1]^2+S[2]^2 && return(S))))}, [1..100])\\ M. F. Hasler, Mar 22 2025

A009129 Perimeter of more than one Pythagorean triangle.

Original entry on oeis.org

60, 84, 90, 120, 132, 144, 168, 180, 210, 240, 252, 264, 270, 280, 288, 300, 312, 330, 336, 360, 390, 396, 408, 420, 432, 440, 450, 456, 462, 468, 480, 504, 510, 520, 528, 540, 546, 552, 560, 570, 576, 588, 600, 612, 616, 624, 630, 660, 672, 680, 684, 690, 700, 720

Offset: 1

David W. Wilson

nonn

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Subsequence of A010814.

is(k)=my(m2,mm2,s); for(m=2,sqrtint(k\2), m2=2*m; mm2=2*m^2; forstep(n=if(m%2,2,1),min(k\m2-m,m-1),m%2+1, if(k%(mm2+m2*n)==0 && gcd(m,n)==1 && s++>1, return(1)))); 0 \\ Charles R Greathouse IV, Feb 03 2018

list(lim)=my(v=List(),ct=Map(),m2,mm2,t); for(m=2, sqrtint(lim\2), m2=2*m; mm2=2*m^2; forstep(n=if(m%2,2,1),min(lim\m2-m,m-1),m%2+1, if(gcd(m,n)>1, next); t=mm2+m2*n; forstep(k=t,lim,t, mapput(ct,k,mapisdefined(ct,k))))); for(n=60,lim, if(mapisdefined(ct,n) && mapget(ct,n), listput(v,n))); ct=0; Vec(v) \\ Charles R Greathouse IV, Feb 03 2018

A334761 Perimeters of Pythagorean triangles whose hypotenuse divides the difference of squares of its long and short legs.

Original entry on oeis.org

60, 120, 180, 240, 300, 360, 390, 420, 480, 540, 600, 660, 680, 720, 780, 840, 900, 960, 1020, 1080, 1140, 1170, 1200, 1260, 1320, 1360, 1380, 1400, 1440, 1500, 1560, 1620, 1680, 1740, 1800, 1860, 1920, 1950, 1980, 2030, 2040, 2100, 2160, 2220, 2280, 2340, 2400

Offset: 1

Wesley Ivan Hurt, May 10 2020

nonn

The smallest terms corresponding to 2,...,5 triangles are a(15) = 780, a(191) = 9360, a(3324) = 159120, and a(19433) = 928200, respectively. - Giovanni Resta, May 11 2020

			a(1) = 60; the triangle [15,20,25] has perimeter 60. The difference of squares of its long and short leg lengths is (20^2 - 15^2) = 400 - 225 = 175 and 25|175.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Wikipedia, Integer Triangle
Wikipedia, Pythagorean Triple.
Index entries related to Pythagorean Triples.

Cf. A005044, A010814.

Reap[Do[s = Solve[x^2 + y^2 == (p-x-y)^2 && z^2 == x^2 + y^2 && 0 0, {x, y, z}, Integers]; If[s != {} && AnyTrue[{x, y , z} /. s, Mod[#[[2]]^2 - #[[1]]^2, #[[3]]] == 0 &], Print@Sow@p], {p, 12, 1000, 2}]][[2, 1]] (* Giovanni Resta, May 11 2020 *)

Terms a(31) and beyond from Giovanni Resta, May 11 2020

A048759 Longest perimeter of a Pythagorean triangle with n as length of one of the three sides.

Original entry on oeis.org

12, 12, 30, 24, 56, 40, 90, 60, 132, 84, 182, 112, 240, 144, 306, 180, 380, 220, 462, 264, 552, 312, 650, 364, 756, 420, 870, 480, 992, 544, 1122, 612, 1260, 684, 1406, 760, 1560, 840, 1722, 924, 1892, 1012, 2070, 1104, 2256, 1200, 2450, 1300, 2652

Offset: 3

Henry Bottomley, Jun 15 2000

Stefano Spezia, Table of n, a(n) for n = 3..10000
Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Cf. A010814, A029578, A055522, A055523, A055524.

[(3*n^2+4*n-n^2*(-1)^n)/4: n in [3..60]]; // Vincenzo Librandi, Jul 19 2015

A048759[n_] := (3 - (-1)^n)*n^2 / 4 + n; Array[A048759, 100, 3] (* or *)
LinearRecurrence[{0, 3, 0, -3, 0, 1}, {12, 12, 30, 24, 56, 40}, 100] (* Paolo Xausa, Feb 29 2024 *)

Vec(-2*x^3*(2*x^5+x^4-6*x^3-3*x^2+6*x+6)/((x-1)^3*(x+1)^3) + O(x^100)) \\ Colin Barker, Sep 13 2014

a(n) = n*A029578(n+2) = n+A055523(n)+A055524(n).
a(2*k) = 2*k*(k+1), a(2*k+1) = 2*(2*k+1)*(k+1).
a(n) = 3*a(n-2)-3*a(n-4)+a(n-6). - Colin Barker, Sep 13 2014
G.f.: -2*x^3*(2*x^5+x^4-6*x^3-3*x^2+6*x+6) / ((x-1)^3*(x+1)^3). - Colin Barker, Sep 13 2014
a(n) = (3*n^2+4*n-n^2*(-1)^n)/4. - Luce ETIENNE, Jul 18 2015
E.g.f.: x*((4 + x)*cosh(x) + (3 + 2*x)*sinh(x) - 4*(1 + x))/2. - Stefano Spezia, May 24 2021

A370599 a(n) is the number of distinct triangles with integral side-lengths for which the perimeter 2*n divides the area.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 0, 1, 0, 0, 1, 0, 0, 2, 0, 1, 1, 1, 2, 2, 0, 0, 1, 1, 0, 4, 0, 1, 3, 0, 0, 1, 0, 0, 0, 0, 0, 3, 1, 2, 1, 0, 0, 3, 0, 0, 3, 2, 1, 4, 0, 2, 0, 3, 0, 2, 0, 0, 2, 2, 3, 1, 0, 2, 4, 1, 0, 8, 1, 0, 1

Offset: 1

Felix Huber, Mar 09 2024

If the perimeter 2*n of a triangle with integral edge-lengths divides its area A, then this also applies to a triangle stretched with positive integer k, because A*k^2/(2*n*k) = k*A/(2*n). Therefore a(d) <= a(n) for all positive divisors d of n and a(m) >= a(n) for all positive integer multiples m of n.

With an odd perimeter, according to Heron's formula the area A would have the form A = sqrt((2*k - 1)/8), where k is a positive integer. The area A would be irrational and the integer perimeter would not divide the area A. For this reason, only triangles with an even perimeter are considered in this sequence.

			a(18) = 1, because only the triangle (9, 10, 17) satisfies the condition: A/(2*n) = 36/36 = 1. (9, 10, 17) is one of the five triangles for which the perimeter is equal to the area (see A098030).
a(42) = 4, because exactly the 4 triangles (10, 35, 39) with A/(2*n) = 168/84 = 2, (14, 30, 40) with A/(2*n) = 168/84 = 2, (15, 34, 35) with A/(2*n) = 252/84 = 3 and (26, 28, 30) with A/(2*n) = 336/84 = 4 satisfy the condition.
a(426) = 0, because no triangle satisfies the condition. Therefore, a(n) = 0 for all n for which n*k = 426 for positive integers k.

Felix Huber, Table of n, a(n) for n = 1..500
Wikipedia, Heron's formula

Cf. A010814, A098030, A221837, A221838, A369951.

A370599 := proc(n) local u, v, w, A, q, i; i := 0; for u to floor(2/3*n) do for v from max(u, floor(n - u) + 1) to floor(n - 1/2*u) do w := 2*n - u - v; A := sqrt(n*(n - u)*(n - v)*(n - w)); if A = floor(A) then q := 1/2*A/n; if q = floor(q) then i := i + 1; end if; end if; end do; end do; return i; end proc;
seq(A370599(n), n = 1 .. 87);

a(n*k) >= a(n) for positive integers k.

A376608 Sides x < y < z of Pythagorean triangles ordered first by increasing perimeter x+y+z, then by shorter leg x.

Original entry on oeis.org

3, 4, 5, 6, 8, 10, 5, 12, 13, 9, 12, 15, 8, 15, 17, 12, 16, 20, 7, 24, 25, 10, 24, 26, 15, 20, 25, 20, 21, 29, 18, 24, 30, 16, 30, 34, 12, 35, 37, 21, 28, 35, 9, 40, 41, 15, 36, 39, 24, 32, 40, 27, 36, 45, 14, 48, 50, 20, 48, 52, 24, 45, 51, 30, 40, 50, 28, 45, 53, 11, 60, 61, 33, 44, 55

Offset: 1

Hugo Pfoertner, Sep 29 2024

			   Triangle
   |  Perimeter
   |       x   y   z
   1  12 [ 3,  4,  5]
   2  24 [ 6,  8, 10]
   3  30 [ 5, 12, 13]
   4  36 [ 9, 12, 15]
   5  40 [ 8, 15, 17]
   6  48 [12, 16, 20]
   7  56 [ 7, 24, 25]
   8  60 [10, 24, 26]
   9  60 [15, 20, 25]
  10  70 [20, 21, 29]

Cf. A010814, A024364, A103605.

A374597 uses this order of sides.

A270417 Number of integer-sided right triangles with semiperimeter n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 3, 0, 0, 1, 0, 0, 2, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 3, 0, 0, 0, 1, 0, 3, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 2, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1

Offset: 1

Henry Bottomley, Mar 16 2016

nonn

Number of positive integer solutions to x*y*(y+z) = n with y and z coprime and of opposite parity and z < y.

Records occur at 1, 6, 30, 60, 120, 210, 360, 420, 840, 1260, 2310, 2520, 4620, 9240, 13860, 27720, 55440, 60060, ... - Antti Karttunen, Sep 25 2018

			a(25)=0 since 2*25 = 50 is not the perimeter of a suitable triangle;
a(30)=2 since 2*30 = 60 = 15+20+25 = 10+24+26;
a(35)=1 since 2*35 = 70 = 20+21+29.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A010814. Nonzero for terms in A005279.

a[n_] := Count[{x, y, z} /. {ToRules[Reduce[x>0 && y>0 && z>0 && zJean-François Alcover, Jun 03 2017 *)

A270417(n) = { my(s=0); fordiv(n,x,fordiv(n/x,y,my(w=n/(x*y)); if((w < 2*y)&&(w>y)&&(w%2)&&(1==gcd(w,y)),s++))); (s); }; \\ (Here z = w-y) - Antti Karttunen, Sep 25 2018

A334791 Perimeters of Pythagorean triangles with squarefree area.

Original entry on oeis.org

12, 30, 70, 84, 132, 182, 260, 340, 374, 390, 420, 462, 476, 494, 510, 598, 646, 782, 798, 870, 966, 1012, 1054, 1254, 1276, 1302, 1334, 1508, 1518, 1612, 1628, 1716, 1804, 1860, 1892, 1924, 2030, 2046, 2132, 2220, 2262, 2310, 2380, 2444, 2460, 2494, 2516, 2542

Offset: 1

Wesley Ivan Hurt, May 10 2020

nonn

The smallest terms corresponding to 2, 3, and 4 triangles are a(32) = 1716, a(1325) = 81510, and a(5027) = 317460, respectively. - Giovanni Resta, May 11 2020

			a(1) = 12; There is one Pythagorean triangle, [3,4,5], with perimeter 12 whose area 3*4/2 = 6 (squarefree).

Giovanni Resta, Table of n, a(n) for n = 1..10000
Wikipedia, Integer Triangle
Index entries related to Pythagorean Triples.

Cf. A010814.

Reap[ Do[s = Solve[ x^2 + y^2 == (p-x-y)^2 && 0 0, {x, y}, Integers]; If[s != {} && AnyTrue[x y/2 /. s, SquareFreeQ], Print@ Sow@ p], {p, 12, 1000, 2}]][[2, 1]] (* Giovanni Resta, May 11 2020 *)

Terms a(39) and beyond from Giovanni Resta, May 11 2020

A334799 Perimeters of Pythagorean triangles whose short leg divides its area.

Original entry on oeis.org

12, 24, 30, 36, 48, 56, 60, 72, 80, 84, 90, 96, 108, 112, 120, 132, 140, 144, 150, 154, 156, 160, 168, 180, 182, 192, 204, 208, 210, 216, 224, 228, 234, 240, 252, 264, 270, 276, 280, 288, 300, 306, 308, 312, 320, 324, 330, 336, 340, 348, 352, 360, 364, 372, 374, 380

Offset: 1

Wesley Ivan Hurt, May 12 2020

nonn

			a(1) = 12; There is one Pythagorean triangle with perimeter 12, [3,4,5] whose area is 6 and 3|6.
a(2) = 24; There is one Pythagorean triangle with perimeter 24, [6,8,10] whose area is 24 and 6|24.

Wikipedia, Integer Triangle
Wikipedia, Pythagorean Triple.
Index entries related to Pythagorean Triples.

Cf. A010814, A334761, A334801.

cp's OEIS Frontend

A156687 Perimeters of Pythagorean triangles that can be constructed in exactly 5 different ways.

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A382268 Numbers k such that a right triangle can be formed from a chain of linked rods of lengths 1, 2, 3, ..., k, with the perimeter equal to the total length.

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A009129 Perimeter of more than one Pythagorean triangle.

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PARI

A334761 Perimeters of Pythagorean triangles whose hypotenuse divides the difference of squares of its long and short legs.

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Extensions

A048759 Longest perimeter of a Pythagorean triangle with n as length of one of the three sides.

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Magma

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PARI

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A370599 a(n) is the number of distinct triangles with integral side-lengths for which the perimeter 2*n divides the area.

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Maple

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A376608 Sides x < y < z of Pythagorean triangles ordered first by increasing perimeter x+y+z, then by shorter leg x.

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A270417 Number of integer-sided right triangles with semiperimeter n.

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