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

1,2

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.

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A033492 A126160 A118962 * A073413 A046681 A259672

Adjacent sequences: A096030 A096031 A096032 * A096034 A096035 A096036

Keyword

nonn,easy

Author

Lekraj Beedassy, Jun 16 2004

Extensions

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

Erroneous comment deleted by Andrew Weimholt, Sep 07 2010

Status

approved