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A024363 Number of primitive Pythagorean triangles with side n. 4
0, 0, 1, 1, 2, 0, 1, 1, 1, 0, 1, 2, 2, 0, 2, 1, 2, 0, 1, 2, 2, 0, 1, 2, 2, 0, 1, 2, 2, 0, 1, 1, 2, 0, 2, 2, 2, 0, 2, 2, 2, 0, 1, 2, 2, 0, 1, 2, 1, 0, 2, 2, 2, 0, 2, 2, 2, 0, 1, 4, 2, 0, 2, 1, 4, 0, 1, 2, 2, 0, 1, 2, 2, 0, 2, 2, 2, 0, 1, 2, 1, 0, 1, 4, 4, 0, 2, 2, 2, 0, 2, 2, 2, 0, 2, 2, 2, 0, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives number of times AUBUC takes value n.

Using Euclidean parameters (x, y) with x > y to generate primitive Pythagorean triples to capture all occurrences of side n, the Mma program below must allow the x parameter to iterate at least (n+1)/2 times. - Frank M Jackson, Jun 12 2017

LINKS

Frank M Jackson, Table of n, a(n) for n = 1..10000

Ron Knott, Pythagorean Triples and Online Calculators

FORMULA

a(n)=0 for n=1 and n=2 (mod 4)=A016825. a(n)=A024361(n)+A024362(n). - Lekraj Beedassy, Dec 01 2003

MATHEMATICA

lst={}; xmax=51; Do[If[GCD[x, y]==1&&OddQ[x+y], AppendTo[lst, Sort@{x^2-y^2, 2 x*y, x^2+y^2}]], {x, xmax}, {y, x}]; BinCounts[Select[Flatten@lst, #<2xmax &], {1, 2(xmax-1), 1}] (* or *)

a[n_] := Block[{x, y, s = List@ ToRules@ Reduce[(x^2-y^2 == n^2 || x^2 + y^2 == n^2) && x>y>0, {x, y}, Integers]}, If[s == {}, 0, Length@ Select[ {x, y} /. s, GCD @@ # == 1 &]]]; Array[a, 99] (* Giovanni Resta, Jun 19 2017 *)

CROSSREFS

Cf. A024361, A024362.

Sequence in context: A117165 A213629 A278522 * A050600 A129691 A280750

Adjacent sequences:  A024360 A024361 A024362 * A024364 A024365 A024366

KEYWORD

nonn

AUTHOR

David W. Wilson

STATUS

approved

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Last modified September 26 01:25 EDT 2017. Contains 292500 sequences.