A096426 - OEIS

%I #12 Aug 29 2024 02:40:09

%S 6,17,31,48,69,93,121,152,187,225,267,312,360,412,468,526,589,655,724,

%T 797,873,953,1036,1122,1212,1306,1403,1503,1607,1715,1826,1940,2058,

%U 2179,2304,2432,2563,2698,2837,2979,3125,3274,3426,3582,3741,3904,4070

%N Floor of the area consecutive odd sided triangles.

%C 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.

%C 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.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LemniscateConstant.html">Lemniscate Constant</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GausssConstant.html">Gauss's Constant</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MagicHexagon.html">Magic Hexagon</a>

%F 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.

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

%t 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 *)

%o (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)", ") ) }

%Y Cf. A096378.

%K nonn

%O 3,1

%A _Cino Hilliard_, Aug 08 2004