A096378 Floor of area of triangle with consecutive integer sides.

0, 2, 6, 9, 14, 20, 26, 34, 42, 51, 61, 72, 84, 96, 109, 124, 139, 155, 172, 190, 208, 228, 248, 269, 291, 314, 338, 363, 388, 415, 442, 470, 499, 529, 560, 591, 624, 657, 691, 727, 762, 799, 837, 875, 915, 955, 996, 1038, 1081, 1125, 1170, 1215, 1261, 1308

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

Sequence in context: A184824 A184836 A327895 * A342426 A217001 A320666

Adjacent sequences: A096375 A096376 A096377 * A096379 A096380 A096381

Keyword

nonn

Author

Cino Hilliard, Aug 04 2004

Status

approved