A331258 -id:A331258 - cp's OEIS frontend

A331257 Numerator of squared radius of inscribed circle of the n-th triangle with integer sides in the list given by A316841. Denominators are A331258.

1, 3, 1, 9, 5, 1, 3, 5, 4, 7, 3, 45, 4, 25, 21, 1, 75, 9, 2, 63, 12, 25, 35, 9, 27, 8, 7, 9, 11, 5, 27, 2, 175, 3, 49, 3, 3, 147, 11, 5, 135, 8, 245, 13, 3, 99, 20, 225, 18, 49, 63, 16, 55, 5, 189, 16, 39, 4, 165, 3, 15, 48, 15, 7, 117, 12, 275, 45, 441, 16

Offset: 1

Hugo Pfoertner, Jan 26 2020

			   n i (A316843)
   | | j (A316844)
   | | | k (A316845)
   | | | |  a(n) this sequence
   | | | |  |  A331258(n)
   | | | |  |  |  rho = sqrt(a(n)/A331258(n))
   1 1 1 1  1 12  0.28868
   2 2 2 1  3 20  0.38730
   3 2 2 2  1  3  0.57735
   4 3 2 2  9 28  0.56695
   5 3 3 1  5 28  0.42258
   6 3 3 2  1  2  0.70711
   7 3 3 3  3  4  0.86603
   8 4 3 2  5 12  0.64550
   9 4 3 3  4  5  0.89443
  10 4 4 1  7 36  0.44096
  11 4 4 2  3  5  0.77460
  12 4 4 3 45 44  1.01130
  13 4 4 4  4  3  1.15470
  14 5 3 3 25 44  0.75378
  15 5 4 2 21 44  0.69085
  16 5 4 3  1  1  1.00000

Cf. A316841, A316843, A316844, A316845, A331258.

rh2(a, b, c)={my(s=(a+b+c)/2); (s-a)*(s-b)*(s-c)/s};
for(i=1, 8, for(j=1, i, for(k=1, j, if(j+k>i, print1(numerator(rh2(i, j, k)), ", ")))))

Radius of inscribed circle of a triangle with sides (a,b,c):

rho(a,b,c) = sqrt((s - a)*(s - b)*(s - c)/s) with s = (a + b + c)/2.

A140246 Decimal expansion of sqrt(15)/6.

Original entry on oeis.org

6, 4, 5, 4, 9, 7, 2, 2, 4, 3, 6, 7, 9, 0, 2, 8, 1, 4, 1, 9, 6, 5, 4, 4, 2, 3, 3, 2, 9, 7, 0, 6, 6, 6, 0, 1, 8, 0, 5, 4, 8, 6, 9, 5, 0, 8, 8, 1, 9, 3, 1, 8, 0, 4, 4, 3, 1, 2, 6, 2, 2, 9, 4, 3, 5, 2, 2, 4, 7, 1, 8, 1, 9, 8, 9, 4, 9, 6, 5, 0, 5, 5, 8, 6, 5, 4, 7, 8, 9, 6, 1, 4, 3, 1, 1, 2, 2, 5, 2, 9, 8, 6, 0, 5, 0

Offset: 0

Rick L. Shepherd, May 14 2008

Inradius of the obtuse scalene triangle with sides of lengths 2, 3 and 4, the scalene triangle with least integer side lengths. Per the Weisstein link, the inradius is the area divided by the semiperimeter.

			0.64549722436790281419654423329706660180548695088193180443126229435224718198...

Daniel Starodubtsev, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Inradius.
Index entries for algebraic numbers, degree 2

Cf. A140239, A140240, A140241, A140242, A140243, A140244, A140245, A140247, A140248, A140249, A088543, A010472.

Equals sqrt(A331257(8)/A331258(8)) (squared inradii of triangles with integer sides).

RealDigits[Sqrt[15]/6,10,120][[1]] (* Harvey P. Dale, Mar 31 2013 *)

PARI
```
sqrt(15)/6
```

sqrt(15)/6 = A010472/6 = 2*A140239/9.

cp's OEIS Frontend

A331257 Numerator of squared radius of inscribed circle of the n-th triangle with integer sides in the list given by A316841. Denominators are A331258.

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