A096033 Difference between leg and hypotenuse in primitive Pythagorean triangles.

1, 2, 8, 9, 18, 25, 32, 49, 50, 72, 81, 98, 121, 128, 162, 169, 200, 225, 242, 288, 289, 338, 361, 392, 441, 450, 512, 529, 578, 625, 648, 722, 729, 800, 841, 882, 961, 968, 1058, 1089, 1152, 1225, 1250, 1352, 1369, 1458, 1521, 1568, 1681, 1682, 1800, 1849

Offset: 1

Author

Lekraj Beedassy, Jun 16 2004

Keywords

• nonn
• easy

Comments

Consists of the odd squares and the halves of the even squares. - Andrew Weimholt, Sep 07 2010

Question: Do we have a(n) mod 2 = A004641(n)? - David A. Corneth, Jan 02 2019

References

• L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 2, p. 170.

Links

• David A. Corneth, Table of n, a(n) for n = 1..10099
• James M. Parks, Computing Pythagorean Triples, arXiv:2107.06891 [math.GM], 2021.
• James M. Parks, On the Curved Patterns Seen in the Graph of PPTs, arXiv:2104.09449 [math.CO], 2021.

Crossrefs

Cf. A001105, A004641, A016754, A094904, A245058.

Programs

Mathematica

nmax = 100;
Union[2 Range[nmax]^2, (2 Range[0, Ceiling[nmax/Sqrt[2]]] + 1)^2] (* Jean-François Alcover, Jan 01 2019 *)

PARI

upto(n) = vecsort(concat(vector((sqrtint(n)+1)\2, i, (2*i-1)^2), vector(sqrtint(n\2), i, 2*i^2))) \\ David A. Corneth, Jan 02 2019

Formula

Union of A001105 (integers of form 2*n^2) and A016754 (the odd squares).

Sum_{n>=1} 1/a(n) = 5*Pi^2/24 = 10 * A245058. - Amiram Eldar, Feb 14 2021

Extensions

Corrected and extended by Matthew Vandermast and Ray Chandler, Jun 17 2004

Erroneous comment deleted by Andrew Weimholt, Sep 07 2010