OFFSET
1,2
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
FORMULA
Given a triangle ABC with sides a, b, base c, height h and x=base of right triangle formed by a and h. Then a^2 = h^2 + x^2, b^2 = h^2 + (c-x)^2, h = sqrt(a^2 - x^2), area = (1/2)hc. Hence x = (a^2 - b^2 + c^2)/2c and so area = (1/4)*sqrt(4*a^2*c^2 - (a^2 - b^2 + c^2)^2).
From Robert Israel, Jan 16 2018: (Start)
a(n) = floor((n+1)*sqrt(3*(n+3)*(n-1))/4).
sqrt(3)*(n^2/4 + n/2) - 2 < a(n) < sqrt(3)*(n^2/4 + n/2). (End)
EXAMPLE
For triangle with sides 3,4,5 area = (1/4)*sqrt(4*9*25 - (9-16+25)^2) = 6. This is the 3,4,5 right triangle with base 3 and height 4. (1/2)*3*4 = 6.
MAPLE
f:= n -> floor((n+1)*sqrt(3*(n+3)*(n-1))/4):
map(f, [$1..100]); # Robert Israel, Jan 16 2018
PROG
(PARI) area(n) = { for(x=1, n, a=x; b=x+1; c=x+2; z=1/4*sqrt(4*a^2*c^2-(c^2+a^2-b^2)^2); print1(floor(z)", ") ) }
CROSSREFS
KEYWORD
nonn
AUTHOR
Cino Hilliard, Aug 04 2004
STATUS
approved