A055595 -id:A055595 - cp's OEIS frontend

A188158 Area A of the triangles such that A and the sides are integers.

6, 12, 24, 30, 36, 42, 48, 54, 60, 66, 72, 84, 90, 96, 108, 114, 120, 126, 132, 144, 150, 156, 168, 180, 192, 198, 204, 210, 216, 234, 240, 252, 264, 270, 288, 294, 300, 306, 324, 330, 336, 360, 378, 384, 390, 396, 408, 420, 432, 456, 462, 468, 480, 486, 504, 510, 522, 528

Offset: 1

Michel Lagneau, Mar 22 2011

nonn

The area A of a triangle whose sides have lengths a, b, and c is given by Heron's formula: A = sqrt(s*(s-a)*(s-b)*(s-c)), where s = (a+b+c)/2. A given area often corresponds to more than one triangle; for example, a(9) = 60 for the triangles (a,b,c) = (6,25,29), (8,17,15), (13,13,10) and (13,13,24).

If only primitive integer triangles (that is, the lengths of the sides are coprime) are considered, then the possible areas are 6 times the terms in A083875. - T. D. Noe, Mar 23 2011

			a(3) = 24 because the area of the triangle whose sides are 4, 15, 13 is given by sqrt(p(p-4)(p-15)(p-13)) = 24, where p = (4 + 15 + 13)/2 = 16.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Triangle
Wikipedia, Heronian triangle

Cf. A007237, A009112, A024153, A024365, A051516, A051584, A051585, A055592, A055593, A055594, A055595.

# storage of areas in T(i)
T:=array(1..4000):nn:=100:k:=1:for a from 1
  to nn do: for b from 1 to nn do: for c from 1 to nn do: p:=(a+b+c)/2 : x:=p*(p-a)*(p-b)*(p-c):   if x>0 then x1:=abs(x):s:=sqrt(x1) :else fi:if s=floor(s) then T[k]:=s:k:=k+1:else
  fi:od:od:od:
# sort of T(i)
for jj from 1 to k-1 do: ii:=jj:for k1 from  ii+1 to k-1 do:if T[ii]>T[k1] then ii:=k1:else fi:od: m:=T[jj]:T[jj]:=T[ii]:T[ii]:=m:od:liste:=convert(T,set):print(liste):
# second program:
isA188158 := proc(A::integer)
    local Asqr, s,a,b,c ;
    Asqr := A^2 ;
    for s in numtheory[divisors](Asqr) do
        if s^2> A then
        for a from 1 to s-1 do
            if modp(Asqr,s-a) = 0 then
                for b from a to s-1 do
                    c := 2*s-a-b ;
                    if s*(s-a)*(s-b)*(s-c) = Asqr then
                        return true ;
                    end if;
                end do:
            end if;
        end do:
        end if;
    end do:
    false ;
end proc:
for n from 3 to 600 do
    if isA188158(n) then
        printf("%d,\n",n) ;
    end if;
end do: # R. J. Mathar, May 02 2018

nn = 528; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[0 < area2 <= nn^2 && IntegerQ[Sqrt[area2]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; Union[lst] (* T. D. Noe, Mar 23 2011 *)

A070086 Areas of integer triangles [A070080(n), A070081(n), A070082(n)], rounded values.

Original entry on oeis.org

0, 1, 2, 1, 2, 3, 2, 3, 4, 4, 4, 2, 4, 4, 6, 5, 6, 7, 3, 5, 5, 7, 8, 6, 7, 8, 9, 3, 6, 6, 9, 7, 10, 11, 7, 9, 10, 11, 12, 4, 6, 8, 10, 8, 12, 12, 14, 8, 10, 12, 13, 12, 15, 16, 4, 7, 9, 12, 10, 14, 10, 15, 16, 17, 9, 12, 13, 15, 14, 17, 18, 19, 5, 8, 10

Offset: 1

Reinhard Zumkeller, May 05 2002

Triangles [A070080(A070142(n)), A070081(A070142(n)), A070082(A070142(n))] have integer areas = a(A070142(k)) = A070149(k).

			[A070080(25), A070081(25), A070082(25)] = [3,5,6] and s = A070083(25)/2 = (3+5+6)/2 = 7: a(25) = sqrt(s*(s-3)*(s-5)*(s-6)) = sqrt(7*(7-3)*(7-5)*(7-6)) = sqrt(7*4*2*1) = sqrt(56) = 7.48331, rounded = 7.

Jean-François Alcover, Table of n, a(n) for n = 1..972
Eric Weisstein's World of Mathematics, Heron's Formula.
Reinhard Zumkeller, Integer-sided triangles

Cf. A051516, A055595, A046131.
The sides are given by A070080, A070081, A070082.
See A135622 for values signifying the precise area and further crossrefs.

m = 50; (* max perimeter *)
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1]& // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]]&];
area[{a_, b_, c_}] := With[{p = (a+b+c)/2}, Sqrt[p(p-a)(p-b)(p-c)] // Round];
area /@ triangles (* Jean-François Alcover, Oct 03 2021 *)

a(n) = sqrt(s*(s-u)*(s-v)*(s-w)), where u=A070080(n), v=A070081(n), w=A070082(n) and s = A070083(n)/2 = (u+v+w)/2.

A046131 Areas of scalene integer Heronian triangles (A046128, A046129, A046130) sorted by increasing c and b.

Original entry on oeis.org

6, 24, 30, 54, 24, 84, 36, 60, 66, 42, 96, 84, 126, 90, 150, 84, 120, 36, 204, 210, 210, 60, 216, 132, 96, 336, 72, 144, 240, 294, 84, 252, 360, 114, 156, 180, 210, 120, 210, 420, 168, 270, 264, 168, 384, 240, 468, 126, 180, 336, 336, 504, 264, 330, 486, 216

Offset: 0

Eric W. Weisstein

nonn

This is the ordering of triangles used for A316841.

Eric Weisstein's World of Mathematics, Heronian Triangle.

The sides are given by A046128, A046129, A046130.
Cf. A055595, A072294, A316841, A316853.
Range of values: A383413.

sideMax = 60; r[c_] := Reap[Do[ p = (a + b + c)/2; red = Reduce[ area > 1 && a < b < c && area^2 == p*(p - a)*(p - b)*(p - c), area, Integers]; If[red =!= False, sol = {a, b, c, area} /. {ToRules[red]}; Sow[sol]], {b, 1, c - 1}, {a, c - b, b - 1}]]; triangles = Flatten[ Reap[ Do[rc = r[c]; If[rc[[2]] =!= {}, Sow[rc[[2, 1]]]], {c, 5, sideMax}]][[2, 1]] , 2]; Sort[ triangles, Which[#1[[3]] < #2[[3]], True, #1[[3]] > #2[[3]], False, #1[[2]] < #2[[2]], True,  #1[[2]] > #2[[2]], False, #1[[1]] <= #2[[1]], True, True, False] &][[All, 4]] (* Jean-François Alcover, Oct 29 2012 *)

A055592 Longest side of congruent triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.

Original entry on oeis.org

5, 6, 8, 10, 12, 13, 13, 15, 15, 15, 16, 17, 17, 17, 18, 20, 20, 20, 21, 21, 24, 24, 24, 25, 25, 25, 25, 26, 26, 26, 26, 28, 29, 29, 30, 30, 30, 30, 30, 30, 30, 32, 34, 34, 34, 35, 35, 35, 36, 36, 37, 37, 37, 37, 37, 39, 39, 39, 39, 39, 39, 40, 40, 40, 40, 40, 40, 40, 41, 41

Offset: 1

Henry Bottomley, May 26 2000

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A055593, A055594, A055595.

max = 41; triangles = Reap[Do[s = (a+b+c)/2; area = Sqrt[s*(s-a)*(s-b)*(s-c)]; If[IntegerQ[area] && area > 0, Sow[{a, b, c, area}]], {a, 1, max}, {b, a, max}, {c, b, max}]][[2, 1]]; A055592 = Sort[triangles, #1[[3]]*max^2 + #1[[2]]*max + #1[[1]] < #2[[3]]* max^2 + #2[[2]]*max + #2[[1]] &][[All, 3]](* Jean-François Alcover, Jun 12 2012 *)

A055593 Second longest side of congruent triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.

Original entry on oeis.org

4, 5, 5, 8, 10, 12, 13, 12, 13, 14, 10, 10, 15, 17, 15, 13, 15, 16, 17, 20, 13, 15, 20, 17, 20, 24, 25, 24, 25, 25, 26, 25, 21, 25, 17, 24, 25, 25, 26, 28, 29, 20, 20, 30, 34, 28, 29, 34, 29, 30, 20, 26, 30, 35, 37, 25, 28, 34, 35, 36, 39, 25, 26, 29, 30, 32, 37, 39, 28, 40

Offset: 1

Henry Bottomley, May 26 2000

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A055592, A055594, A055595.

max = 41; triangles = Reap[Do[s = (a+b+c)/2; area = Sqrt[s*(s-a)*(s-b)*(s-c)]; If[IntegerQ[area] && area > 0, Sow[{a, b, c, area}]], {a, 1, max}, {b, a, max}, {c, b, max}]][[2, 1]]; A055593 = Sort[triangles, #1[[3]]*max^2 + #1[[2]]*max + #1[[1]] < #2[[3]]* max^2 + #2[[2]]*max + #2[[1]] &][[All, 2]](* Jean-François Alcover, Jun 12 2012 *)

A055594 Shortest side of congruent triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.

Original entry on oeis.org

3, 5, 5, 6, 10, 5, 10, 9, 4, 13, 10, 9, 8, 16, 15, 11, 7, 12, 10, 13, 13, 15, 20, 12, 15, 7, 14, 10, 3, 17, 20, 17, 20, 6, 17, 18, 11, 25, 8, 26, 5, 20, 18, 16, 32, 21, 8, 15, 25, 30, 19, 15, 13, 12, 24, 16, 17, 25, 10, 15, 30, 25, 22, 29, 14, 24, 13, 25, 15, 9, 17, 18, 29, 20, 35

Offset: 1

Henry Bottomley, May 26 2000

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A055592, A055593, A055595.

max = 42; triangles = Reap[Do[s = (a+b+c)/2; area = Sqrt[s*(s-a)*(s-b)*(s-c)]; If[IntegerQ[area] && area > 0, Sow[{a, b, c, area}]], {a, 1, max}, {b, a, max}, {c, b, max}]][[2, 1]]; A055594 = Sort[triangles, #1[[3]]*max^2 + #1[[2]]*max + #1[[1]] < #2[[3]]* max^2 + #2[[2]]*max + #2[[1]] &][[All, 1]](* Jean-François Alcover, Jun 12 2012 *)

A072294 Areas of primitive Heronian triangles sorted by longest side, then by middle side and finally shortest side.

Original entry on oeis.org

6, 12, 12, 30, 60, 24, 84, 36, 60, 120, 66, 42, 84, 126, 60, 90, 84, 168, 36, 204, 210, 210, 60, 120, 132, 72, 84, 252, 360, 114, 156, 180, 210, 420, 120, 210, 420, 168, 420, 240, 468, 126, 180, 336, 360, 420, 264, 330, 252, 168, 504, 420, 780, 420, 306, 456, 156

Offset: 1

Lekraj Beedassy, Jul 12 2002

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 2197 terms from Zak Seidov)
A. Dendane, Heron's Formula for Area of a Triangle-Geometry Calculator
Zak Seidov, Table of n=1..6093, A072294(n), A120131(n), A120132(n), A120133(n) (with A120131(n) <=1000)
Michael Somos, Heronian Triangle Table
P. Yiu, Heron triangles with sides < 100 Appendix Chap. 9.3 pp. 81/360 in 'Recreational Mathematics'.

Cf. A120131, A120132, A120133, A055595.

nn = 200; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s] && GCD[a, b, c] == 1, area2 = s (s - a) (s - b) (s - c); If[area2 > 0 && IntegerQ[Sqrt[area2]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a}, {c, b}]; lst (* T. D. Noe, Mar 23 2011 *)

More terms from Ray Chandler, Jul 02 2004

A070786 Area of triangles with sides whose squares are integers and with positive integer area, ordered by longest side, then second longest side and finally shortest side.

Original entry on oeis.org

1, 1, 2, 2, 1, 3, 1, 2, 3, 4, 3, 2, 1, 3, 3, 6, 2, 4, 2, 4, 6, 1, 3, 5, 2, 4, 3, 6, 2, 3, 1, 5, 1, 4, 4, 2, 6, 7, 3, 6, 8, 6, 3, 1, 3, 5, 6, 4, 8, 2, 5, 10, 1, 4, 2, 6, 3, 7, 9, 5, 9, 7, 3, 2, 4, 1, 8, 9, 5, 7, 10, 3, 6, 9, 12, 4, 2, 8, 6, 4, 2, 10, 8, 12, 6, 2, 6, 3, 1, 7, 5, 10, 11, 4, 8, 7, 14, 3, 6, 3

Offset: 1

Henry Bottomley, May 07 2002

nonn

			a(6)=3 since the triangle with sides sqrt(9), sqrt(8) and sqrt(5) has area 3.

A. Bogomolny, Sam Loyd's Geometric Puzzle

Cf. A055595, A070783, A070784, A070785, A070787.

A068966 Areas of integer Heronian triangles [prime(A068964(n)), prime(A068964(n)+1), A068965(n)].

Original entry on oeis.org

6, 66, 3204, 6810, 37716, 72006, 182430, 532236, 370614, 2155134, 3203694, 6353634, 51712890, 42020844, 28698786, 33163770, 55637466, 125033580, 172985436, 105470250, 151375626, 178631034, 185921166, 217064574, 939603126, 376267326, 1812742734, 2193232470, 853918566

Offset: 1

Reinhard Zumkeller, Mar 30 2002

nonn

			A068964(3) = 24: [prime(24), prime(25), A068965(3)] = [89,97,170], with s = (89+97+170)/2 = 178: Area^2 = s*(s-89)*(s-81)*(s-170) = 178*89*81*8 = 10265616 = 3204*3204, therefore a(3) = 3204.

Giovanni Resta, Table of n, a(n) for n = 1..385
Eric Weisstein's World of Mathematics, Heronian Triangle.
Eric Weisstein's World of Mathematics, Heron's Formula.

Cf. A046131, A055595, A068969.

Terms a(13) and beyond from Giovanni Resta, Apr 20 2020

A068969 Areas of integer Heronian triangles [A068967(n), prime(A068967(n)), A068968(n)].

Original entry on oeis.org

6, 210, 528, 1680, 1800, 5304, 6930, 3150, 4650, 21000, 32760, 69342, 53550, 2170560, 2200200, 2501070, 646800, 4777080, 4796550, 11865840, 12243840, 12863760, 18064200, 30510480, 3232320, 66023100, 70691400, 47977380, 144357720, 185560830, 156128700, 320843040

Offset: 1

Reinhard Zumkeller, Mar 30 2002

nonn

			a(5) = 37: [37, A000040(37), A068968(5)] = [37,157,130], with s = (36+157+130)/2 = 162: Area^2 = s*(s-37)*(s-157)*(s-130) = 162*125*5*32 = 3240000 = 1800*1800, therefore a(5) = 1800.

Giovanni Resta, Table of n, a(n) for n = 1..120
Eric Weisstein's World of Mathematics, Heronian Triangle.
Eric Weisstein's World of Mathematics, Heron's Formula.

Cf. A046131, A055595, A068966.

area[{a_, b_, c_}] := Block[{s = (a + b + c)/2}, Sqrt[s (s-a) (s-b) (s-c)]]; zz[n_] := Block[{s, p = Prime[n], t, z, A}, s = Solve[(n^2 - 2 n p + p^2 - z) (z - n^2 - 2 n p - p^2) == t^2 && z>0 && t>0, {t, z}, Integers]; If[s == {}, {}, z = Sort@ Select[Sqrt[z /. s], IntegerQ]]; Select[Table[{n, p, e}, {e, z}], IntegerQ[A = area[#]] && A > 0 &]]; area /@ Join @@ Parallelize@ Array[zz, 4300] (* Giovanni Resta, Apr 20 2020 *)

Erroneous term 25070627 removed and more terms from Giovanni Resta, Apr 20 2020

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A188158 Area A of the triangles such that A and the sides are integers.

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A070086 Areas of integer triangles [A070080(n), A070081(n), A070082(n)], rounded values.

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A046131 Areas of scalene integer Heronian triangles (A046128, A046129, A046130) sorted by increasing c and b.

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Mathematica

A055592 Longest side of congruent triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.

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Mathematica

A055593 Second longest side of congruent triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.

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Mathematica

A055594 Shortest side of congruent triangles with integer sides and positive integer area, ordered by longest side, then second longest side and finally shortest side.

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Mathematica

A072294 Areas of primitive Heronian triangles sorted by longest side, then by middle side and finally shortest side.

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A070786 Area of triangles with sides whose squares are integers and with positive integer area, ordered by longest side, then second longest side and finally shortest side.

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A068966 Areas of integer Heronian triangles [prime(A068964(n)), prime(A068964(n)+1), A068965(n)].

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