A135622 -id:A135622 - cp's OEIS frontend

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

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.

A317182 Numbers k such that k = 16*area(T)^2 for an integer triangle, T.

Original entry on oeis.org

3, 15, 35, 48, 63, 99, 128, 135, 143, 195, 231, 240, 243, 255, 275, 320, 323, 351, 384, 399, 455, 483, 495, 560, 575, 576, 663, 675, 735, 768, 783, 819, 855, 896, 899, 935, 975, 1008, 1023, 1071, 1155, 1215, 1235, 1280, 1295, 1311, 1344, 1443, 1463, 1536, 1539

Offset: 1

N. J. A. Sloane, Jul 25 2018

nonn

The possible lengths of the chord connecting the cusps of the lens-like intersection area between two circles with integer radii and integer distance d between their centers are of the form sqrt(a(n))/d. - Hugo Pfoertner, Sep 05 2020

Lars Blomberg, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Circle-Circle Intersection

Cf. A316841.

Sorted and uniqued values of A135622 or A316853.

More terms from Lars Blomberg, Apr 25 2019

Name edited by Peter Munn, Jul 30 2025

A331011 16 * squared area of triangles with integer sides i <= j <= k, such that more than one ordered triple of sides produces the same area.

Original entry on oeis.org

63, 495, 675, 768, 1008, 1071, 1280, 1575, 2304, 2499, 2835, 2880, 2975, 3135, 3456, 3591, 4095, 4275, 4455, 4608, 4928, 5103, 5760, 5775, 6615, 6656, 6831, 6975, 7040, 7488, 7875, 7920, 8064, 8415, 8448, 8463, 8775, 8855, 8960, 9135, 9216, 9600, 9984, 10535

Offset: 1

Hugo Pfoertner, Jan 06 2020

nonn

			a(1) = 63: triangles (1,4,4) and (2,2,3) both have squared area 3.9375 = 63/16. All smaller squared area values A(i,j,k) correspond to unique triples of side lengths: A(1,1,1) = 0.1875 = 3/16, A(1,2,2) = 0.9375 = 15/16, A(1,3,3) = 2.1875 = 35/16, A(2,2,2) = 3 = 48/16.
a(2) = 495: A(2,6,7) = A(3,4,4) = 30.9375 = 495/16.
a(8) = 1575: A(2,11,12) = A(3,8,10) = A(4,5,6) = 98.4375 = 1575/16.
a(23) = 5760: A(2,19,19) = A(3,13,14) = A(6,7,7) = A(6,7,11) = 360 = 5760/16.

Subset of A317182.
Nonunique terms of A135622 or A316853.
Cf. A331012.

A371973 a(n) is the number of distinct areas > 0 of triangles with integer sides and perimeter n.

Original entry on oeis.org

1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, 12, 16, 13, 19, 14, 21, 19, 23, 20, 27, 23, 30, 27, 32, 29, 35, 32, 39, 34, 44, 39, 48, 43, 52, 47, 55, 51, 60, 53, 63, 59, 69, 58, 74, 67, 78, 73, 84, 75, 90, 81, 92, 88, 101, 91, 108, 93, 112, 106

Offset: 3

Hugo Pfoertner, Apr 16 2024

nonn

Hugo Pfoertner, Table of n, a(n) for n = 3..10000

Cf. A005044, A007237, A024153, A317182, A371070.

See the formula section for the relationships with A026810, A070083, A135622 (which has many crossrefs related to areas of triangles).

A2(a,b,c) = {my (s=(a+b+c)/2); s*(s-a)*(s-b)*(s-c)};
a371973(n) = {my (A=List()); forpart (v=n, listput(A, A2(v[1],v[2],v[3])), [1,(n-1)\2], [3,3]); #Set(A)};

def A371973(n): return len(set((2*(b+c)-n)*(n-2*b)*(n-2*c) for c in range((n+2)//3, (n+1)//2) for b in range((n-c+1)//2, c+1))) # David Radcliffe, Aug 01 2025

a(n) = |{A135622(k) : A070083(k) = n}| = |{A135622(k) : A026810(n) < k <= A026810(n+1)}|. - Peter Munn, Jul 29 2025

b-file corrected by David Radcliffe, Aug 01 2025

cp's OEIS Frontend

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

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A317182 Numbers k such that k = 16*area(T)^2 for an integer triangle, T.

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A331011 16 * squared area of triangles with integer sides i <= j <= k, such that more than one ordered triple of sides produces the same area.

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A371973 a(n) is the number of distinct areas > 0 of triangles with integer sides and perimeter n.

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