A096426 Floor of the area consecutive odd sided triangles.

6, 17, 31, 48, 69, 93, 121, 152, 187, 225, 267, 312, 360, 412, 468, 526, 589, 655, 724, 797, 873, 953, 1036, 1122, 1212, 1306, 1403, 1503, 1607, 1715, 1826, 1940, 2058, 2179, 2304, 2432, 2563, 2698, 2837, 2979, 3125, 3274, 3426, 3582, 3741, 3904, 4070

Offset

3,1

Comments

The area of an odd sided triangle is irrational. Proof: Area = (1/4)*sqrt((c+b-a)*(a-c+b)*(a+c-b)*(a+c+b)) The sides of an odd sided triangle are of the form 4k+1 or 4k+3.

All permutations of the remainders of sides 4k+r for the factors (c+b-a),(a-c+b),(a+c-b),(a+b+c) evaluate to 1 1 1 3 or 3 3 3 1 Thus the remainder of D ==(c+b-a)*(a-c+b)*(a+c-b)*(a+c+b) mod 4 is 3 => D is not square => Area is irrational.

Links

Table of n, a(n) for n=3..49.

Eric Weisstein's World of Mathematics, Lemniscate Constant

Eric Weisstein's World of Mathematics, Gauss's Constant

Eric Weisstein's World of Mathematics, Magic Hexagon

Formula

Area = (1/4)*sqrt((c+b-a)*(a-c+b)*(a+c-b)*(a+c+b)) where a < b < c are the sides of a triangle. Floor(Area) is this sequence.

Example

Triangle with sides 3,5,7 units has area = 6.4951905283832..sq units. 6 is the first entry in the table.

Mathematica

fa[n_]:=Module[{s=Total[n]/2}, Floor[Sqrt[s(s-n[[1]])(s-n[[2]])(s- n[[3]])]]]; fa/@Partition[Range[3, 101, 2], 3, 1] (* Harvey P. Dale, Apr 25 2011 *)

Prog

(PARI) area(n) = { for(x=1, n, a=x+x+1; b=a+2; c=b+2; y=1/4*sqrt((c+b-a)*(a-c+b)*(a+c-b)*(a+c+b)); print1(floor(y)", ") ) }

Crossrefs

Cf. A096378.

Sequence in context: A212461 A204059 A067559 * A130051 A338894 A237658

Adjacent sequences: A096423 A096424 A096425 * A096427 A096428 A096429

Keyword

nonn

Author

Cino Hilliard, Aug 08 2004

Status

approved