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A096426
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Floor of the area consecutive odd sided triangles.
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0
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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
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OFFSET
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3,1
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COMMENTS
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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.
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LINKS
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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
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FORMULA
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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.
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EXAMPLE
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Triangle with sides 3,5,7 units has area = 6.4951905283832..sq units. 6 is the
first entry in the table.
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MATHEMATICA
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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] (* From Harvey P. Dale, Apr 25 2011 *)
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PROG
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(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)", ") ) }
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CROSSREFS
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Cf. A096378.
Sequence in context: A212461 A204059 A067559 * A130051 A038795 A216892
Adjacent sequences: A096423 A096424 A096425 * A096427 A096428 A096429
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KEYWORD
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nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)gmail.com), Aug 08 2004
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STATUS
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approved
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