A051516 -id:A051516 - cp's OEIS frontend

A330916 Sum of the largest side lengths of all Heronian triangles with perimeter A051518(n).

5, 6, 8, 10, 13, 27, 61, 17, 35, 20, 59, 41, 96, 25, 80, 139, 30, 26, 57, 157, 37, 37, 140, 296, 40, 196, 207, 250, 209, 91, 587, 52, 294, 51, 267, 214, 498, 50, 539, 117, 310, 697, 530, 147, 206, 342, 503, 856, 73, 744, 75, 68, 85, 550, 793, 256, 172, 155, 1270, 1202

Offset: 1

Wesley Ivan Hurt, May 02 2020

nonn

			a(1) = 5; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its largest side length is 5.
a(6) = 27; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12]. The sum is 15 + 12 = 27.

Eric Weisstein's World of Mathematics, Heronian Triangle
Wikipedia, Heronian triangle
Wikipedia, Integer Triangle

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.

Cf. A330912, A330915.

a(n) = Sum_{k=1..floor(c(n)/3)} Sum_{i=k..floor((c(n)-k)/2)} sign(floor((i+k)/(c(n)-i-k+1))) * chi(sqrt((c(n)/2)*(c(n)/2-i)*(c(n)/2-k)*(c(n)/2-(c(n)-i-k)))) * (c(n)-i-k), where chi(n) = 1 - ceiling(n) + floor(n) and c(n) = A051518(n). - Wesley Ivan Hurt, May 12 2020

A070139 Number of isosceles integer triangles with perimeter n having integral area.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A051516(n) - A024153(n).

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Heronian Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A051516, A024153, A059169, A070145.

A070209 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an integer triangle with integer inradius.

Original entry on oeis.org

17, 116, 212, 269, 368, 370, 493, 561, 587, 659, 850, 1204, 1297, 1582, 1599, 1629, 1920, 1988, 2115, 2352, 2555, 2574, 2774, 2778, 3251, 3473, 3746, 3751, 4286, 4298, 4307, 4313, 4319, 4330, 4370, 4406, 5008, 5251

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

			a(3)=212: [A070080(212), A070081(212), A070082(212)] = [5,12,13], for s = A070083(212)/2 = (5+12+13)/2 = 15: inradius = sqrt((s-5)*(s-12)*(s-13)/s) = sqrt(10*3*2/15) = sqrt(4) = 2; therefore A070200(212)=2. [Corrected by _Rick L. Shepherd_, May 15 2008]

Mohammad K. Azarian, Circumradius and Inradius, Problem S125, Math Horizons, Vol. 15, Issue 4, April 2008, p. 32. Solution published in Vol. 16, Issue 2, November 2008, p. 32.

Eric Weisstein's World of Mathematics, Incircle.
R. Zumkeller, Integer-sided triangles

Cf. A070142, A070201, A051516.

A070140 Number of acute integer triangles with perimeter n having integral area.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A051516(n) - A070141(n) - A024155(n).

Eric Weisstein's World of Mathematics, Heronian Triangle.
Eric Weisstein's World of Mathematics, Acute Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A051516, A070093, A070146.

A330921 Sum of the areas of all Heronian triangles with perimeter A051518(n).

Original entry on oeis.org

6, 12, 12, 24, 30, 72, 198, 60, 126, 66, 288, 180, 360, 84, 330, 648, 132, 204, 420, 876, 114, 156, 840, 1764, 264, 1350, 1632, 2016, 1830, 624, 3816, 330, 2604, 456, 2280, 2352, 4800, 780, 4422, 1224, 2940, 7068, 5430, 912, 2310, 3744, 5520, 9144, 984, 8736, 1020

Offset: 1

Wesley Ivan Hurt, May 02 2020

nonn

			a(1) = 6; there is one Heronian triangle with perimeter A051518(1) = 12, which is [3,4,5] and its area is 3*4/2 = 6.
a(6) = 72; there are two Heronian triangles with perimeter A051518(6) = 32, [4,13,15] and [10,10,12]. The sum of their areas 24 + 48 = 72.

Eric Weisstein's World of Mathematics, Heronian Triangle
Wikipedia, Heronian triangle
Wikipedia, Integer Triangle

Cf. A051516, A051518, A070138, A096468, A298079, A298614, A305717.

Cf. A330912, A330915, A330916.

a(n) = Sum_{k=1..floor(c(n)/3)} Sum_{i=k..floor((c(n)-k)/2)} sign(floor((i+k)/(c(n)-i-k+1))) * chi(sqrt((c(n)/2)*(c(n)/2-i)*(c(n)/2-k)*(c(n)/2-(c(n)-i-k)))) * sqrt((c(n)/2)*(c(n)/2-i)*(c(n)/2-k)*(c(n)/2-(c(n)-i-k))), where chi(n) = 1 - ceiling(n) + floor(n) and c(n) = A051518(n). - Wesley Ivan Hurt, May 12 2020

A334665 Perimeters of Heronian triangles with mutually distinct side lengths.

Original entry on oeis.org

12, 24, 30, 32, 36, 40, 42, 44, 48, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 84, 88, 90, 96, 98, 100, 104, 108, 110, 112, 114, 120, 126, 128, 130, 132, 136, 140, 144, 150, 152, 154, 156, 160, 162, 164, 168, 170, 172, 174, 176, 180, 182, 186, 190, 192, 196, 198, 200, 204

Offset: 1

Wesley Ivan Hurt, May 07 2020

nonn

			a(1) = 12; there is one Heronian triangle with perimeter 12 such that all side lengths are mutually distinct, [3,4,5].
a(4) = 32; there is one Heronian triangle with perimeter 32 such that all side lengths are mutually distinct, [4,13,15].

Eric Weisstein's World of Mathematics, Heronian Triangle
Wikipedia, Heronian triangle
Wikipedia, Integer Triangle

Cf. A051516, A051518.

A385736 a(n) is the number of distinct nondegenerate triangles with perimeter n whose side lengths are triangular numbers.

Original entry on oeis.org

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

Offset: 0

Felix Huber, Jul 16 2025

0, 1, 6, 10, 28, 55 are the only triangular numbers <= 10^6 that are not perimeters of triangles whose side lengths are triangular numbers. Conjecture: There are no other triangular numbers that have this property.

			The a(31) = 2 distinct nondegenerate triangles with perimeter 31 and whose side lengths are triangular numbers are [1, 15, 15] and [6, 10, 15].

Felix Huber, Table of n, a(n) for n = 0..10000
Felix Huber, Maple program to compute the counted triangles
Eric Weisstein's World of Mathematics, Triangular Number

Cf. A000217, A005044, A051516, A070088, A366398, A380875, A385737, A385860, A385872.

A385736:=proc(N) # To get the first N + 1 terms.
    local p,x,y,z,i;
    p:=[];
    for z to floor((sqrt(24*N+9)-3)/6) do
        for x from z to floor((sqrt(4*N-3)-1)/2) do
            for y from max(z,floor((sqrt(1+4*(x^2+x-z^2-z))-1)/2)+1) to min(x,floor((sqrt(1+4*(2*N-x^2-x-z^2-z))-1)/2)) do
                p:=[op(p),z*(z+1)/2+y*(y+1)/2+x*(x+1)/2]
            od
        od
    od;
    return seq(numboccur(p,i),i=0..N)
end proc;
A385736(87);

Trivial upper bound: a(n) <= A005044(n).

a(A385737(n)) >= 1.

A070141 Number of obtuse integer triangles with perimeter n having integral area.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, May 05 2002

nonn

a(n) = A051516(n) - A070140(n) - A024155(n).

Eric Weisstein's World of Mathematics, Heronian Triangle.
Eric Weisstein's World of Mathematics, Obtuse Triangle.
R. Zumkeller, Integer-sided triangles

Cf. A051516, A070101, A070147.

A331366 Number of Heronian triangles with mutually distinct side lengths and perimeter A334665(n).

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, May 07 2020

nonn

			a(1) = 1; there is one Heronian triangle with perimeter A334665(1) = 12 such that all side lengths are mutually distinct, [3,4,5].
a(4) = 1; there is one Heronian triangle with perimeter A334665(4) = 32 such that all side lengths are mutually distinct, [4,13,15].

Eric Weisstein's World of Mathematics, Heronian Triangle
Wikipedia, Heronian triangle
Wikipedia, Integer Triangle

Cf. A051516, A051518, A334665.

A385737 Perimeters of nondegenerate triangles with integer areas, whose side lengths are triangular numbers.

Original entry on oeis.org

176, 224, 264, 336, 504, 644, 756, 950, 1196, 1232, 1280, 1500, 1566, 1650, 1700, 2100, 2112, 2250, 2366, 2754, 3036, 3306, 5676, 5796, 7296, 8064, 8316, 8526, 9576, 10206, 10260, 12474, 13200, 15872, 16236, 16896, 17094, 17150, 20172, 21714, 21726, 22382, 22644

Offset: 1

Felix Huber, Jul 16 2025

nonn

224 and 1280 are the only perimeters <= 10^6 of nondegenerate triangles whose side lengths (28, 91, 105 or 325, 325, 630, respectively) and areas (1176 or 25200, respectively) are triangular numbers.

			176 is a term because it is the perimeter of the triangle [55, 55, 66], where 55 and 66 are triangular numbers, which has an integer area of sqrt(88*(88 - 55)*(88 - 55)*(88 - 66)) = 1452.
224 is a term because it is the perimeter of the triangle [28, 91, 105], where 28, 91 and 105 are triangular numbers, which has an integer area of sqrt(112*(112 - 28)*(112 - 91)*(112 - 105)) = 1176 (which is also a triangular number).

Felix Huber, Table of n, a(n) for n = 1..315
Felix Huber, Maple program to compute the triangles (incl. areas) with perimeter a(n)
Eric Weisstein's World of Mathematics, Triangular Number

Subsequence of A380875.

Cf. A000217, A051516, A070083, A385736, A385872.

A385737:=proc(P) # To get all perimeters <= P.
    local p,x,y,z,u,v,w,s;
    p:=[];
    for z to floor((sqrt(24*P+9)-3)/6) do
        for x from z to floor((sqrt(4*P-3)-1)/2) do
            for y from max(z,floor((sqrt(1+4*(x^2+x-z^2-z))-1)/2)+1) to min(x,floor((sqrt(1+4*(2*P-x^2-x-z^2-z))-1)/2)) do
            	u:=z*(z+1)/2;
            	v:=y*(y+1)/2;
            	w:=x*(x+1)/2;
            	s:=(u+v+w)/2;
            	if issqr(s*(s-u)*(s-v)*(s-w)) then
               	    p:=[op(p),u+v+w]
               	fi
            od
        od
    od;
    return op(sort(p))
end proc;
A385737(22644);

cp's OEIS Frontend

A330916 Sum of the largest side lengths of all Heronian triangles with perimeter A051518(n).

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A070139 Number of isosceles integer triangles with perimeter n having integral area.

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A070209 Numbers n such that [A070080(n), A070081(n), A070082(n)] is an integer triangle with integer inradius.

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A070140 Number of acute integer triangles with perimeter n having integral area.

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A330921 Sum of the areas of all Heronian triangles with perimeter A051518(n).

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A334665 Perimeters of Heronian triangles with mutually distinct side lengths.

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A385736 a(n) is the number of distinct nondegenerate triangles with perimeter n whose side lengths are triangular numbers.

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Maple

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A070141 Number of obtuse integer triangles with perimeter n having integral area.

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A331366 Number of Heronian triangles with mutually distinct side lengths and perimeter A334665(n).

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A385737 Perimeters of nondegenerate triangles with integer areas, whose side lengths are triangular numbers.

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