A024361 Number of primitive Pythagorean triangles with leg n.

0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 0, 2, 1, 1, 0, 1, 2, 2, 0, 1, 2, 1, 0, 1, 2, 1, 0, 1, 1, 2, 0, 2, 2, 1, 0, 2, 2, 1, 0, 1, 2, 2, 0, 1, 2, 1, 0, 2, 2, 1, 0, 2, 2, 2, 0, 1, 4, 1, 0, 2, 1, 2, 0, 1, 2, 2, 0, 1, 2, 1, 0, 2, 2, 2, 0, 1, 2, 1, 0, 1, 4, 2, 0, 2, 2, 1, 0, 2, 2, 2, 0, 2, 2, 1, 0, 2, 2, 1, 0, 1, 2, 4

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

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

For n > 1, a(n) = 0 for n == 2 (mod 4) (n in A016825).

From Jianing Song, Apr 23 2019: (Start)

Note that all the primitive Pythagorean triangles are given by A = min{2*u*v, u^2 - v^2}, B = max{2*u*v, u^2 - v^2}, C = u^2 + v^2, where u, v are coprime positive integers, u > v and u - v is odd. As a result:

(a) if n is odd, then a(n) is the number of representations of n to the form n = u^2 - v^2, where u, v are coprime positive integers (note that this guarantees that u - v is odd) and u > v. Let s = u + v, t = u - v, then n = s*t, where s and t are unitary divisors of n and s > t, so the number of representations is A034444(n)/2 if n > 1 and 0 if n = 1;

(b) if n is divisible by 4, then a(n) is the number of representations of n to the form n = 2*u*v, where u, v are coprime positive integers (note that this also guarantees that u - v is odd because n/2 is even) and u > v. So u and v must be unitary divisors of n/2, so the number of representations is A034444(n/2)/2. Since n is divisible by 4, A034444(n/2) = A034444(n) so a(n) = A034444(n)/2.

(c) if n == 2 (mod 4), then n/2 is odd, so n = 2*u*v implies that u and v are both odd, which is not acceptable, so a(n) = 0.

a(n) = 0 if n = 1 or n == 2 (mod 4), otherwise a(n) is a power of 2.

The earliest occurrence of 2^k is 2*A002110(k+1) for k > 0. (End)

Examples

			a(12) = 2 because 12 appears twice, in (A,B,C) = (5,12,13) and (12,35,37).
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000
• Ron Knott, Pythagorean Triples and Online Calculators
• J. S. Myers, R. Schroeppel, S. R. Shannon, N. J. A. Sloane, and P. Zimmermann, Three Cousins of Recaman's Sequence, arXiv:2004:14000 [math.NT], April 2020.
• Amitabha Tripathi, On Pythagorean triples containing a fixed integer, Fibonacci Quart. 46/47 (2008/09), no. 4, 331-340.
• Eric Weisstein's World of Mathematics, Pythagorean Triple

Crossrefs

Cf. A024359, A024360, A024362, A024363, A046079, A020883, A020884, A034444.

Programs

Mathematica

Table[If[n == 1 || Mod[n, 4] == 2, 0, 2^(Length[FactorInteger[n]] - 1)], {n, 100}]

PARI

A024361(n) = if(1==n||(2==(n%4)),0,2^(omega(n)-1)); \\ (after the Mathematica program) - Antti Karttunen, Nov 10 2018

Formula

a(n) = A034444(n)/2 = 2^(A001221(n)-1) if n != 2 (mod 4) and n > 1, a(n) = 0 otherwise. - Jianing Song, Apr 23 2019

a(n) = A024359(n) + A024360(n). - Ray Chandler, Feb 03 2020

Extensions

Incorrect comment removed by Ant King, Jan 28 2011

More terms from Antti Karttunen, Nov 10 2018