A120062
Number of triangles with integer sides a <= b <= c having integer inradius n.
Original entry on oeis.org
1, 5, 13, 18, 15, 45, 24, 45, 51, 52, 26, 139, 31, 80, 110, 89, 33, 184, 34, 145, 185, 103, 42, 312, 65, 96, 140, 225, 36, 379, 46, 169, 211, 116, 173, 498, 38, 123, 210, 328, 44, 560, 60, 280, 382, 134, 64, 592, 116, 228, 230, 271, 47, 452, 229, 510, 276, 134, 54
Offset: 1
a(1)=1: {3,4,5} is the only triangle with integer sides and inradius 1.
a(2)=5: {5,12,13}, {6,8,10}, {6,25,29}, {7,15,20}, {9,10,17} are the only triangles with integer sides and inradius 2.
a(4)=A120252(1)+A120252(2)+A120252(4)=1+4+13 because 1, 2 and 4 are the factors of 4. The 1 primitive triangle with inradius n=1 is (3,4,5). The 4 primitive triangles with n=2 are (5,12,13), (9,10,17), (7,15,20), (6,25,29). The 13 primitive triangles with n=4 are (13,14,15), (15,15,24), (11,25,30), (15,26,37), (10,35,39), (9,40,41), (33,34,65), (25,51,74), (9,75,78), (11,90,97), (21,85,104), (19,153,170), (18,289,305). (Primitive means GCD(a, b, c, n)=1.)
Cf.
A078644 [Pythagorean triangles with inradius n],
A057721 [n^4+3*n^2+1].
Let S(n) be the set of triangles with integer sides a<=b<=c and inradius n. Then:
A120062(n) gives number of triangles in S(n).
A120261(n) gives number of triangles in S(n) with gcd(a, b, c) = 1.
A120252(n) gives number of triangles in S(n) with gcd(a, b, c, n) = 1.
A005408(n) = 2n+1 gives shortest short side a of triangles in S(n).
A120064(n) gives shortest middle side b of triangles in S(n).
A120063(n) gives shortest long side c of triangles in S(n).
A120570(n) gives shortest perimeter of triangles in S(n).
A120572(n) gives smallest area of triangles in S(n).
A058331(n) = 2n^2+1 gives longest short side a of triangles in S(n).
A082044(n) = n^4+2n^2+1 gives longest middle side b of triangles in S(n).
A057721(n) = n^4+3n^2+1 gives longest long side c of triangles in S(n).
A120571(n) = 2n^4+6n^2+4 gives longest perimeter of triangles in S(n).
A120573(n) = gives largest area of triangles in S(n).
Cf.
A120252 [primitive triangles with integer inradius],
A120063 [minimum of longest sides],
A057721 [maximum of longest sides],
A120064 [minimum of middle sides],
A082044 [maximum of middle sides],
A005408 [minimum of shortest sides],
A058331 [maximum of shortest sides],
A007237 [number of triangles with integer sides and area = n times perimeter].
A120063
Shortest side c of all integer-sided triangles with sides a<=b<=c and inradius n.
Original entry on oeis.org
5, 10, 12, 15, 25, 24, 35, 30, 36, 39, 55, 45, 65, 63, 53, 60, 85, 68, 95, 75, 77, 88, 115, 85, 125, 130, 108, 105, 145, 106, 155, 120, 132, 170, 137, 135, 185, 190, 156, 150, 205, 154, 215, 165, 159, 230, 235, 170, 245, 195, 204, 195, 265, 204, 200, 195, 228, 290
Offset: 1
a(1)=5 because the only triangle with integer sides and inradius 1 is {3,4,5}; its longest side is 5.
a(2)=10: The triangles with inradius 2 are {5,12,13}, {6,8,10}, {6,25,29}, {7,15,20}, {9,10,17}. The minimum of their longest sides is min(13,10,29,20,17)=10.
- 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.
See
A120062 for sequences related to integer-sided triangles with integer inradius n.
Cf.
A120062 [triangles with integer inradius],
A120252 [primitive triangles with integer inradius],
A057721 [maximum of longest sides],
A058331 [maximum of shortest sides],
A120064 [minimum of middle sides],
A082044 [maximum of middle sides],
A005408 [minimum of shortest sides],
A007237.
A120064
Shortest side b of all integer-sided triangles with sides a<=b<=c and inradius n.
Original entry on oeis.org
4, 8, 10, 14, 20, 20, 28, 28, 30, 39, 44, 40, 52, 56, 50, 56, 68, 60, 76, 70, 70, 87, 92, 80, 100, 100, 90, 97, 116, 100, 124, 112, 110, 136, 120, 120, 148, 152, 130, 140, 164, 140, 172, 154, 150, 184, 188, 160, 196, 174, 170, 182, 212, 180, 196, 189, 190, 232, 236
Offset: 1
a(1)=2 because the only triangle with integer sides a<=b<c and inradius 1 is {3,4,5}; its middle side is 4.
a(2)=8: The triangles with inradius 2 are {5,12,13}, {6,8,10}, {6,25,29}, {7,15,20}, {9,10,17}. The minimum of their middle sides is min(12,8,25,15,10)=8.
- 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.
Cf.
A120062 [triangles with integer inradius],
A120252 [primitive triangles with integer inradius],
A057721 [maximum of longest sides],
A120063 [minimum of longest sides],
A058331 [maximum of shortest sides],
A082044 [maximum of middle sides],
A005408 [minimum of shortest sides],
A007237.
See
A120062 for sequences related to integer-sided triangles with integer inradius n.
A120261
Number of primitive triangles with integer sides a<=b<=c and inradius n; primitive means gcd(a, b, c) = 1.
Original entry on oeis.org
1, 4, 10, 11, 13, 28, 17, 26, 31, 31, 20, 77, 28, 46, 67, 40, 28, 100, 26, 72, 120, 62, 32, 139, 44, 53, 71, 118, 32, 202, 35, 70, 135, 73, 97, 211, 33, 80, 130, 134, 36, 284, 45, 141, 183, 78, 50, 226, 68, 112, 150, 146, 38, 173, 150, 219, 182, 80, 38, 468, 36, 82
Offset: 1
a(3)=10 because 10 triangles have coprime integer sides and inradius 3, namely (7,24,25) (7,65,68) (8,15,17) (11,13,20) (12,55,65) (13,40,51) (15,28,41) (16,25,39) (19,20,37) (11,100,109).
- 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.
See
A120062 for sequences related to integer-sided triangles with integer inradius n.
Showing 1-4 of 4 results.
Comments