A362670 Integer inradii for which there exists an isosceles triangle with integer sides (a, a, c) where a < c.

3, 4, 6, 8, 9, 12, 15, 16, 18, 20, 21, 24, 27, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 51, 52, 54, 56, 57, 60, 63, 64, 66, 68, 69, 70, 72, 75, 76, 78, 80, 81, 84, 87, 88, 90, 92, 93, 96, 99, 100, 102, 104, 105, 108, 111, 112, 114, 116, 117, 120, 123, 124, 126, 128, 129, 132, 135

Offset

1,1

Comments

The inradius for isosceles triangle (a, a, c) is r = (c/2)*sqrt((2*a-c)/(2*a+c)).

If m is a term, so is k*m with k > 1.

As r = 3 and r = 4 are terms, A008585 and A008586 are respective subsequences; the only terms < 100 that are not multiples of 3 or 4 are 35 and 70, the next one is r = 154 = 2*7*11 for triple (765, 765, 1386).

By the triangle inequality, a+1 <= c <= 2*a-1.

Differs from A059267. Examples: 154 is not in A059267 but in this sequence at radius r=154 with side lengths c=1386 and a=765. 442 is not in A059267 but in this sequences with r=442, c=6630, a=3435. - R. J. Mathar, Jun 26 2023

Links

Table of n, a(n) for n=1..69.

R. J. Mathar, Solution strategy and Maple program

Eric Weisstein's World of Mathematics, Incircle.

Eric Weisstein's World of Mathematics, Isosceles Triangle.

Example

The smallest inradius r = 3 corresponds to isosceles triangle (10, 10, 12).
The second inradius r = 4 corresponds to isosceles triangle (15, 15, 24).
r = 15 is the first inradius for which there exist two such isosceles triangles: (50, 50, 60) and (68, 68, 120).
r = 35 is the smallest inradius that is not multiple of 3 or of 4, this inradius corresponds to isosceles triangle (222, 222, 420).

Crossrefs

Cf. A008585, A008586, A070204, A120062, A120570.

Cf. A362669 (similar but with (a,b,b)).

Sequence in context: A284469 A366288 A059267 * A355200 A049433 A250984

Adjacent sequences: A362667 A362668 A362669 * A362671 A362672 A362673

Keyword

nonn

Author

Bernard Schott, May 05 2023

Status

approved