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A046079 Number of Pythagorean triangles with leg n. 20
0, 0, 1, 1, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 4, 3, 1, 2, 1, 4, 4, 1, 1, 7, 2, 1, 3, 4, 1, 4, 1, 4, 4, 1, 4, 7, 1, 1, 4, 7, 1, 4, 1, 4, 7, 1, 1, 10, 2, 2, 4, 4, 1, 3, 4, 7, 4, 1, 1, 13, 1, 1, 7, 5, 4, 4, 1, 4, 4, 4, 1, 12, 1, 1, 7, 4, 4, 4, 1, 10, 4, 1, 1, 13, 4, 1, 4, 7, 1, 7, 4, 4, 4, 1, 4, 13, 1, 2, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,8

COMMENTS

Number of ways in which n can be the leg (other than the hypotenuse) of a primitive or nonprimitive right triangle.

Number of ways that 2/n can be written as a sum of exactly two distinct unit fractions. For every solution to 2/n = 1/x + 1/y, x < y, the Pythagorean triple is (n, y-x, x+y-n). - T. D. Noe, Sep 11 2002

For n>2, the positions of the ones in this sequence correspond to the prime numbers and their doubles, A001751 - Ant King, Jan 29 2011

Let L = length of longest leg, H = hypotenuse. For odd n: L =(n^2-1)/2 and H = L+1.  For even n,  L = (n^2-4)/4 and H = L+2. - Richard R. Forberg, May 31 2013.

REFERENCES

A. Beiler, Recreations in the Theory of Numbers. New York: Dover, pp. 116-117, 1966.

LINKS

Table of n, a(n) for n=1..99.

H. Becker, Pythagorean triples triplets in JavaScript

Ron Knott, Pythagorean Triples and Online Calculators

Project Euler, Problem 176: Rectangular triangles that share a cathetus.

F. Richman, Pythagorean Triples

Eric Weisstein's World of Mathematics, Pythagorean Triple

FORMULA

For odd n, a(n) = A018892(n) - 1.

Let n = (2^a0)*(p1^a1)*...*(pk^ak). Then a(n) = [(2*a0 - 1)*(2*a1 + 1)*(2*a2 + 1)*(2*a3 + 1)*...*(2*ak + 1) - 1]/2. Note that if there is no a0 term, i.e. if n is odd, then the first term is simply omitted. - Temple Keller (temple.keller(AT)gmail.com), Jan 05 2008

For odd n, a(n) = (tau(n^2) - 1) / 2; for even n, a(n) = (tau((n / 2)^2) - 1) / 2. - Amber Hu (hupo001(AT)gmail.com), Jan 23 2008

MATHEMATICA

f[n_] := (DivisorSigma[0, If[OddQ[n], n, n / 2]^2] - 1) / 2; Table[f[i], {i, 100}] (* Amber Hu (hupo001(AT)gmail.com), Jan 23 2008 *)

PROG

(Sage) def A046079(n) : return (number_of_divisors(n^2 if n%2==1 else n^2/4) - 1) // 2 # Eric M. Schmidt, Jan 26 2013

CROSSREFS

Cf. A046080, A046081, A001227, A018892, A024361, A024362, A024363.

Sequence in context: A157654 A078692 A033151 * A165509 A100996 A232504

Adjacent sequences:  A046076 A046077 A046078 * A046080 A046081 A046082

KEYWORD

nonn

AUTHOR

Eric W. Weisstein

STATUS

approved

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Last modified October 21 23:13 EDT 2014. Contains 248381 sequences.