A331040
Numerator of squared radius of inscribed circle of a triangle with integer sides i <= j <= k, such that the number of triangles with this radius sets a new record. Denominators are A331041.
Original entry on oeis.org
1, 35, 3, 7, 3, 15, 8, 35, 55, 63, 95, 119, 135, 56, 231, 255, 80, 351, 455, 495, 855, 216, 224, 1071, 360
Offset: 1
b(1) = a(1)/A331041(1) = 1/12: Triangle (1,1,1) has the least possible radius of incircle = sqrt(1/12).
b(2) = a(2)/A331041(2) = 35/52: Triangles (2,18,19) and (3,4,6) are the first occurrence of more than one triangle with the same radius of their incircles. rho = sqrt(35/52) in both cases.
b(3) = a(3)/A331041(3) = 3/4: Triangles are (2,7,7), (3,3,3), and (3,5,7).
b(4) = a(4)/A331041(4) = 7/4: (3,12,12), (3,22,23), (4,5,6), (5,18,22), (6,11,16) are the A331043(4) = 5 triangles with rho^2 = b(4).
b(15) = 231/4 includes the rare case, where two distinct integer solutions for the same pair of sides a and b exist: (20,37,38) and (20,37,39), both with rho^2=231/4 and thus contributing 2 of the A331043(15)=84 triangles with this squared radius of the incircle.
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\\ Only suitable for demonstration of initial terms
rh2(a,b,c)={my(s=(a+b+c)/2);(s-a)*(s-b)*(s-c)/s};
lim_a(x)=ceil(4*(x^2+2));
lim_b(x)=ceil(4*(x^4+2*x^2+1));
target=35/4; v=vector(333938); n=0;
for(a=1,lim_a(sqrt(target)), for(b=a,lim_b(sqrt(target)), for(c=b,a+b-1, f=rh2(a,b,c);v[n++]=f)));
v=vecsort(v); print("A331040 A331041 A331043"); print(numerator(v[1])," ",denominator(v[1])," ",1); m=0; mm=0; for(k=2,#v, if(v[k]>target,break); if(v[k]==v[k-1], m++; if(m>mm&&v[k+1]>v[k], print(numerator(v[k])," ",denominator(v[k])," ",m);mm=m),m=1));
A331043
a(n) is the number of triangles with integer sides i <= j <= k with squared radius of incircle b(n) = A331040(n)/A331041(n). Records of numbers of distinct triangles such that all smaller radii produce fewer triangles sharing the same radius of incircle than the current radius b(n).
Original entry on oeis.org
1, 2, 3, 5, 6, 13, 14, 20, 24, 42, 45, 50, 68, 72, 84, 88, 101, 120, 149, 175, 181, 189, 206, 243, 289
Offset: 1
A331042
a(n) = 4 * squared radius of inscribed circles of triangles with integer sides i <= j <= k, such that the number of triangles with this radius sets a new record. If this radius is not a multiple of (1/4), a(n) = 0.
Original entry on oeis.org
0, 0, 3, 7, 12, 15, 32, 35, 55, 63, 95, 119, 135, 224, 231, 255, 320, 351, 455, 495, 855, 864, 896, 1071, 1440
Offset: 1
A331224
Numerator of squared radius of circumscribed circle of a triangle with integer sides i <= j <= k, such that the number of triangles with this radius sets a new record. Denominators are A331225.
Original entry on oeis.org
1, 64, 49, 1024, 2025, 4096, 25600, 2401, 7744, 148225, 8281, 2073600, 123904, 774400, 3705625
Offset: 1
Correspondence of the first terms b(n) = a(n)/A331225(n) with triangles (i, j, k):
b(1) = 1/3: (1,1,1), start with 1 = A331226(1) triangle.
b(2) = 64/15: (2,3,4), (2,4,4) is the first occurrence of 2 = A331226(2) triangles with identical R.
b(3) = 49/3: (3,5,7), (3,7,8), (5,7,8), (7,7,7) is the first occurrence of more triangles with identical R than the previous record 2, new record is 4 = A331226(3).
b(4) = 1024/15: (5,8,12), (5,14,16), (8,8,14), (8,12,16), (8,16,16), (12,14,16) is the first occurrence of more triangles with identical R than the previous record 4, new record is 6 = A331226(4).
A331225
Denominator of squared radius of circumscribed circle of a triangle with integer sides i <= j <= k, such that the number of triangles with this radius sets a new record. Numerators are A331224.
Original entry on oeis.org
3, 15, 3, 15, 11, 15, 39, 3, 7, 96, 3, 119, 7, 39, 96
Offset: 1
A331226
a(n) is the number of triangles with integer sides i <= j <= k with squared radius of circumscribed circle b(n) = A331224(n)/A331225(n). Records of numbers of distinct triangles such that all smaller radii produce fewer triangles sharing the same radius of circumcircle than the current radius b(n).
Original entry on oeis.org
1, 2, 4, 6, 8, 9, 10, 13, 17, 22, 31, 33, 46, 53, 67
Offset: 1
Showing 1-6 of 6 results.
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