A316853 Areas of all nondegenerate integer triangles, T, expressed as 16*area(T)^2. a(n) is for the triangle with sides A316841(n, 1..3).

3, 15, 48, 63, 35, 128, 243, 135, 320, 63, 240, 495, 768, 275, 231, 576, 975, 99, 384, 819, 1344, 1875, 455, 1008, 351, 896, 1575, 2304, 143, 560, 1215, 2048, 2975, 3888, 735, 675, 1536, 2499, 495, 1280, 2295, 3456, 4655, 195, 768, 1683, 2880, 4275

Offset

1,1

Comments

The squared area of an integer triangle is necessarily a multiple of 1/16.

Links

Lars Blomberg, Table of n, a(n) for n = 1..3000

Eric Weisstein's World of Mathematics, Triangle Area.

Formula

a(n) = 16*s*(s-a)*(s-b)*(s-c) where s = A316851(n)/2, and a,b,c are A316843(n), A316844(n), A316845(n).

Crossrefs

Cf. A316841 (table of triangle sides), A316843 (longest sides), A316844 (middle sides), A316845 (shortest sides), A316851 (perimeters).

Range of values: A317182.

A135622 is the equivalent sequence with the triangles ordered by perimeter.

A317181 is the subsequence for primitive triangles (as listed in A316842).

The subsequence for triangles with integer area is A055595 (whose supersequence A070786 is notably relevant).

Sequence in context: A296799 A328111 A135622 * A061316 A124161 A089580

Adjacent sequences: A316850 A316851 A316852 * A316854 A316855 A316856

Keyword

nonn

Author

N. J. A. Sloane, Jul 23 2018

Extensions

Name edited by Peter Munn, May 10 2025

Status

approved