A342720 -id:A342720 - cp's OEIS frontend

A342721 a(n) is the number of concave integer quadrilaterals (up to congruence) with integer side lengths a,b,c,d with n=Max(a,b,c,d), integer diagonals e,f and integer area.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 0, 2, 0, 0, 0, 3, 1, 1, 0, 0, 1, 3, 0, 0, 0, 2, 1, 0, 6, 0, 4, 4, 2, 1, 0, 0, 1, 0, 0, 6, 0, 2, 8, 6, 2, 0, 1, 2, 0, 2, 0, 9, 0, 0, 2, 0, 13, 1, 0, 4, 0, 3, 0, 3, 5, 10, 11

Offset: 1

Herbert Kociemba, Mar 19 2021

nonn

Without loss of generality we assume that a is the largest side length and that the diagonal e divides the concave quadrilateral into two triangles with sides a,b,e and c,d,e. Then e < a is a necessary condition for concavity. The triangle inequality further implies e > a-b and abs(e-c) < d < e+c.

			a(66)=1 because the only concave integer quadrilateral with longest edge length 66 and integer area has sides a=66, b=55, c=12, d=65, diagonals e=55, f=65 and area 1650.

Cf. A340858 for trapezoids, A342720 for concave integer quadrilaterals with arbitrary area.

an={};
area[a_,b_,c_,d_,e_,f_]:=(1/4)Sqrt[(4e^2 f^2-(a^2+c^2-b^2-d^2)^2)]
he[a_,b_,e_]:=(1/(2 e))Sqrt[(-((a-b-e) (a+b-e) (a-b+e) (a+b+e)))];
paX[e_]:={e,0} (*vertex A coordinate*)
pbX[a_,b_,e_]:={(-a^2+b^2+e^2)/(2 e),he[a,b,e]}(*vertex B coordinate*)
pc={0,0};(*vertex C coordinate*)
pdX[c_,d_,e_]:={(c^2-d^2+e^2)/(2 e),-he[c,d,e]}(*vertex D coordinate*)
concaveQ[{bx_,by_},{dx_,dy_},e_]:=If[by dx-bx dy<0||by dx-bx dy>(by-dy) e,True,False]
gQ[x_,y_]:=Module[{z=x-y,res=False},Do[If[z[[i]]>0,res=True;Break[],If[z[[i]]<0,Break[]]],{i,1,4}];res]
canonicalQ[{a_,b_,c_,d_}]:=Module[{m={a,b,c,d}},If[(gQ[{b,a,d,c},m]||gQ[{d,c,b,a},m]||gQ[{c,d,a,b},m]),False,True]]
Do[cnt=0;
Do[pa=paX[e];pb=pbX[a,b,e];pd=pdX[c,d,e];
If[(f=Sqrt[(pb-pd).(pb-pd)];IntegerQ[f])&&(ar=area[a,b,c,d,e,f]; IntegerQ[ar])&&concaveQ[pb,pd,e]&&canonicalQ[{a,b,c,d}],cnt++
(*;Print[{{a,b,c,d,e,f,ar},Graphics[Line[{pa,pb,pc,pd,pa}]]}]*)],
{b,1,a},{e,a-b+1,a-1},{c,1,a},{d,Abs[e-c]+1,Min[a,e+c-1]}];
AppendTo[an,cnt],{a,1,75}
]
an

A342722 a(n) is the number of convex integer quadrilaterals (up to congruence) with integer side lengths a,b,c,d with n=Max(a,b,c,d) and integer diagonals e,f.

Original entry on oeis.org

0, 0, 0, 2, 2, 1, 5, 7, 8, 5, 7, 13, 14, 11, 15, 31, 18, 14, 18, 30, 25, 24, 22, 64, 42, 35, 51, 58, 34, 48, 37, 87, 71, 46, 69, 74, 51, 53, 74, 110, 53, 72, 61, 96, 106, 73, 60, 181, 102, 103, 125, 134, 79, 118, 133, 215, 141, 82, 82, 221

Offset: 1

Herbert Kociemba, Apr 25 2021

nonn

Without loss of generality we assume that a is the largest side length and that the diagonal e divides the convex quadrilateral into two triangles with sides a,b,e and c,d,e. The triangle inequality implies e > a-b and abs(e-c) < d < e+c.

			a(6)=1 because the only convex integer quadrilateral with longest edge length 6 is a trapezoid with sides a=6, b=5, c=4, d=5 and diagonals e=f=7.

Cf. A340858 for trapezoids, A342720 and A342721 for concave integer quadrilaterals, A342723 for convex integer quadrilaterals with integer area.

an={};
he[a_,b_,e_]:=1/(2 e) Sqrt[-(a-b-e) (a+b-e) (a-b+e) (a+b+e)];
paX[e_]:={e,0} (*vertex A coordinate*)
pbX[a_,b_,e_]:={(-a^2+b^2+e^2)/(2 e),he[a,b,e]}(*vertex B coordinate*)
pc={0,0};(*vertex C coordinate*)
pdX[c_,d_,e_]:={(c^2-d^2+e^2)/(2 e),-he[c,d,e]}(*vertex D coordinate*)
convexQ[{bx_,by_},{dx_,dy_},e_]:=If[(by-dy) e>by dx-bx dy>0,True,False]
(*define order on tuples*)
gQ[x_,y_]:=Module[{z=x-y,res=False},Do[If[z[[i]]>0,res=True;Break[],If[z[[i]]<0,Break[]]],{i,1,6}];res]
(*check if tuple is canonical*)
canonicalQ[{a_,b_,c_,d_,e_,f_}]:=Module[{x={a,b,c,d,e,f}},If[(gQ[{b,a,d,c,e,f},x]||gQ[{d,c,b,a,e,f},x]||gQ[{c,d,a,b,e,f},x]||gQ[{b,c,d,a,f,e},x]||gQ[{a,d,c,b,f,e},x]||gQ[{c,b,a,d,f,e},x]||gQ[{d,a,b,c,f,e},x]),False,True]]
Do[cnt=0;
Do[pa=paX[e];pb=pbX[a,b,e];pd=pdX[c,d,e];
If[(f=Sqrt[(pb-pd).(pb-pd)];IntegerQ[f])&&convexQ[pb,pd,e]&&canonicalQ[{a,b,c,d,e,f}],cnt++
(*;Print[{{a,b,c,d,e,f},Graphics[Line[{pa,pb,pc,pd,pa}]]}]*)],
{b,1,a},{e,a-b+1,a+b-1},{c,1,a},{d,Abs[e-c]+1,Min[a,e+c-1]}];
AppendTo[an,cnt],{a,1 ,60}
]
an

A342723 a(n) is the number of convex integer quadrilaterals (up to congruence) with integer side lengths a,b,c,d with n=Max(a,b,c,d), integer diagonals e,f and integer area.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 2, 1, 1, 3, 1, 2, 0, 0, 4, 3, 0, 0, 4, 7, 2, 0, 5, 2, 5, 0, 1, 0, 3, 4, 3, 4, 0, 6, 7, 3, 4, 0, 3, 4, 0, 0, 5, 0, 9, 10, 9, 3, 0, 5, 8, 0, 4, 0, 17, 4, 0, 9, 1, 19, 2, 0, 6, 2, 7, 0, 7, 7, 7, 23, 2, 8, 12, 0, 10, 0, 5, 0, 15, 27

Offset: 1

Herbert Kociemba, Apr 25 2021

nonn

Without loss of generality we assume that a is the largest side length and that the diagonal e divides the convex quadrilateral into two triangles with sides a,b,e and c,d,e. The triangle inequality implies e > a-b and abs(e-c) < d < e+c.

			a(4)=1 is the smallest possible solution and is a rectangle with a=c=4, b=d=3, e=f=5 and area 12. a(24)=4 includes the smallest possible solution with all sides a,b,c,d different and a=24, b=20, c=15, d=7, e=20, f=25 and area 234. Furthermore there are three rectangles with a=24,b=7, a=24,b=10 and a=24,b=18.

Cf. A340858 for trapezoids, A342720 and A342721 for concave integer quadrilaterals, A342722 for convex integer quadrilaterals with arbitrary area.

an={};
area[a_,b_,c_,d_,e_,f_]:=1/4 Sqrt[4e^2 f^2-(a^2+c^2-b^2-d^2)^2];
he[a_,b_,e_]:=1/(2 e) Sqrt[-(a-b-e) (a+b-e) (a-b+e) (a+b+e)];
paX[e_]:={e,0} (*vertex A coordinate*)
pbX[a_,b_,e_]:={(-a^2+b^2+e^2)/(2 e),he[a,b,e]}(*vertex B coordinate*)
pc={0,0};(*vertex C coordinate*)pdX[c_,d_,e_]:={(c^2-d^2+e^2)/(2 e),-he[c,d,e]}(*vertex D coordinate*)
convexQ[{bx_,by_},{dx_,dy_},e_]:=If[(by-dy) e>by dx-bx dy>0,True,False]
(*define order on tuples*)
gQ[x_,y_]:=Module[{z=x-y,res=False},Do[If[z[[i]]>0,res=True;Break[],If[z[[i]]<0,Break[]]],{i,1,6}];res]
(*check if tuple is canonical*)
canonicalQ[{a_,b_,c_,d_,e_,f_}]:=Module[{x={a,b,c,d,e,f}},If[(gQ[{b,a,d,c,e,f},x]||gQ[{d,c,b,a,e,f},x]||gQ[{c,d,a,b,e,f},x]||gQ[{b,c,d,a,f,e},x]||gQ[{a,d,c,b,f,e},x]||gQ[{c,b,a,d,f,e},x]||gQ[{d,a,b,c,f,e},x]),False,True]]
Do[cnt=0;
Do[pa=paX[e];pb=pbX[a,b,e];pd=pdX[c,d,e];
If[(f=Sqrt[(pb-pd).(pb-pd)];IntegerQ[f])&&(ar=area[a,b,c,d,e,f]; IntegerQ[ar])&&convexQ[pb,pd,e]&&canonicalQ[{a,b,c,d,e,f}],cnt++
(*;Print[{{a,b,c,d,e,f,ar},Graphics[Line[{pa,pb,pc,pd,pa}]]}]*)],{b,1,a},{e,a-b+1,a+b-1},{c,1,a},{d,Abs[e-c]+1,Min[a,e+c-1]}];
AppendTo[an,cnt],{a,1,85}]
an

A344528 a(n) is the number of integer quadrilaterals (up to congruence) with integer side lengths a,b,c,d with n = Max(a,b,c,d) and integer diagonals e,f.

Original entry on oeis.org

0, 0, 0, 2, 2, 1, 5, 7, 8, 5, 7, 13, 14, 11, 16, 31, 20, 14, 19, 33, 26, 26, 24, 67, 43, 39, 53, 62, 36, 53, 40, 94, 72, 48, 73, 77, 64, 60, 94, 122, 58, 75, 68, 106, 109, 81, 62, 195, 114, 113, 140, 151, 87, 129, 143, 235, 154, 97, 92, 266

Offset: 1

Herbert Kociemba, May 22 2021

nonn

This sequence is the sum of the sequences A342720 and A342722 which deal with concave and convex quadrilaterals respectively.

			a(6)=1 because the only integer quadrilateral with longest edge length 6 is a trapezoid with sides a=6, b=5, c=4, d=5 and diagonals e=f=7.

Cf. A342720, A342721, A342722, A342723, A344529.

a(n) = A342720(n) + A342722(n).

A344529 a(n) is the number of integer quadrilaterals (up to congruence) with integer side lengths a,b,c,d with n = Max(a,b,c,d), integer diagonals e,f and integer area.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 2, 1, 1, 4, 1, 4, 0, 0, 6, 3, 0, 0, 7, 8, 3, 0, 5, 3, 8, 0, 1, 0, 5, 5, 3, 10, 0, 10, 11, 5, 5, 0, 3, 5, 0, 0, 11, 0, 11, 18, 15, 5, 0, 6, 10, 0, 6, 0, 26, 4, 0, 11, 1, 32, 3, 0, 10, 2, 10, 0, 10, 12, 17, 34

Offset: 1

Herbert Kociemba, May 22 2021

nonn

This sequence is the sum of the sequences A342721 and A342723 which deal with concave and convex quadrilaterals respectively.

			a(4)=1 because the smallest possible quadrilateral is a rectangle with a=c=4, b=d=3, e=f=5 and area 12.

Cf. A342720, A342721, A342722, A342723, A344528.

a(n) = A342721(n) + A342723(n).

A371969 Perimeters of triangles with integer sides, which can be decomposed into 3 triangles that have a common vertex strictly inside the surrounding triangle and also integer sides.

Original entry on oeis.org

49, 50, 54, 64, 75, 78, 80, 88, 90, 91, 98, 100, 104, 108, 112, 117, 120, 121, 125, 126, 128, 133, 136, 140, 144, 147, 150, 156, 160, 162, 165, 168, 169, 170, 175, 176, 180, 182, 184, 188, 192, 195, 196, 198, 200, 203, 208, 210, 216, 220, 224, 225, 231, 234, 238, 240

Offset: 1

Klaus Nagel and Hugo Pfoertner, Apr 14 2024

nonn

			a(1) = 49 is the perimeter of the first decomposable triangle with sides of the outer triangle [8, 19, 22], and sides meeting at the 4th "inner" vertex: 17, 6, 4. The 3 inner triangles have sides [8, 4, 6], [19, 17, 4], and [22, 6, 17].

These triangles can be viewed as degenerate tetrahedrons, in which all triangular inequalities for the faces are satisfied, and the Cayley-Menger determinant, which determines whether the 4th vertex yields a valid tetrahedron, takes the value 0.

Klaus Nagel, Illustration of a(1) = 49.
Klaus Nagel, Illustration of a(2) = 50.
Klaus Nagel, Illustration of a(3) = 54.
Klaus Nagel, Illustration of a(4) = 64; another triangle is [20, 21, 23].
Hugo Pfoertner, List of triangles up to perimeter 400; one example of each.
Wikipedia, Cayley-Menger determinant

Cf. A070083, A316841, A342720, A371345, A371973.

H(a,b,c) = {my (s=(a+b+c)/2); s*(s-a)*(s-b)*(s-c)};
CM(w1,w2,w3,v1,v2,v3) = matdet([0,1,1,1,1; 1,0,w3^2,w2^2,v1^2; 1,w3^2,0,w1^2,v2^2; 1,w2^2,w1^2,0,v3^2; 1,v1^2,v2^2,v3^2,0]);
is_a371969(peri) = {forpart (w=peri, my (A=H(w[1],w[2],w[3]), epsA=1e-12); for (v1=1, w[3]-2, for (v2=w[3]-v1+1, w[3]-2, my (A3=H(w[3],v2,v1)); if (A3>=A, next); for (v3=1, w[3]-2, if (v3+v2<=w[1] || v3+v1<=w[2], next); my (A1=H(w[1],v2,v3)); if (A1>=A, next); my (A2=H(w[2],v1,v3)); if (A2>=A, next); my (C=CM(w[1],w[2],w[3],v1,v2,v3)); if (C==0 && abs(sqrt(A)-sqrt(A1)-sqrt(A2)-sqrt(A3)) < epsA,
\\ print (peri," ",Vec(w)," ",[v1,v2,v3]);
return(1))))), [1,(peri-1)\2], [3,3]); 0};
for (n=3, 100, if (is_a371969(n), print1(n,", ")))

cp's OEIS Frontend

A342721 a(n) is the number of concave integer quadrilaterals (up to congruence) with integer side lengths a,b,c,d with n=Max(a,b,c,d), integer diagonals e,f and integer area.

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A342722 a(n) is the number of convex integer quadrilaterals (up to congruence) with integer side lengths a,b,c,d with n=Max(a,b,c,d) and integer diagonals e,f.

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A342723 a(n) is the number of convex integer quadrilaterals (up to congruence) with integer side lengths a,b,c,d with n=Max(a,b,c,d), integer diagonals e,f and integer area.

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A344528 a(n) is the number of integer quadrilaterals (up to congruence) with integer side lengths a,b,c,d with n = Max(a,b,c,d) and integer diagonals e,f.

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A344529 a(n) is the number of integer quadrilaterals (up to congruence) with integer side lengths a,b,c,d with n = Max(a,b,c,d), integer diagonals e,f and integer area.

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A371969 Perimeters of triangles with integer sides, which can be decomposed into 3 triangles that have a common vertex strictly inside the surrounding triangle and also integer sides.

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