A096377 Floor of area of triangle with consecutive prime sides.
0, 6, 12, 38, 71, 107, 158, 218, 317, 436, 550, 696, 817, 961, 1184, 1425, 1666, 1883, 2134, 2377, 2635, 3008, 3437, 3931, 4351, 4645, 4887, 5199, 5778, 6548, 7484, 7955, 8653, 9237, 10032, 10642, 11389, 12150, 12928, 13653, 14570, 15323, 16232, 16683
Offset: 1
Keywords
Examples
For triangle with sides 3,5,7 area = (1/4)*sqrt(4*9*49 - (9-25+49)^2) = 6.495...
Programs
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PARI
area(n) = { for(x=1,n, a=prime(x);b=prime(x+1);c=prime(x+2); z=1/4*sqrt(4*a^2*c^2-(c^2+a^2-b^2)^2); print1(floor(z)",") ) }
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 = 1/2hc. 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).