A094904 -id:A094904 - cp's OEIS frontend

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

Lekraj Beedassy, Jun 16 2004

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

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.

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.

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

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

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

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

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

Erroneous comment deleted by Andrew Weimholt, Sep 07 2010

A321067 Considering Pythagorean triple (a,b,c) with a < b < c, least a such that there exists a primitive triple where c - b is the n-th term of A096033.

Original entry on oeis.org

3, 8, 20, 33, 48, 65, 88, 119, 140, 204, 207, 252, 297, 336, 396, 429, 540, 555, 616, 696, 731, 832, 893, 1036, 1113, 1140, 1248, 1311, 1428, 1525, 1692, 1748, 1809, 1960, 2059, 2184, 2325, 2508, 2576, 2739, 2832

Offset: 1

Jeffrey Burch, Oct 26 2018

nonn

			a(2) = 8 because in the primitive triple (8,15,17), c - b = A096033(2) = 2 and no smaller a yields a primitive triple where a < b < c and c - b = 2.

Jean-François Alcover, Table of n, a(n) for n = 1..1000

Cf. A096033, A094904.

nmax = 100; kmax = 2;
A096033 = Union[2 Range[nmax]^2, (2 Range[0, Ceiling[nmax/Sqrt[2]]]+1)^2];
r[n_, k_] := Module[{a, b, c}, {a, b, c} /. {ToRules[Reduce[0 < a < b < c && c - b == A096033[[n]] && a^2 + b^2 == c^2, {a, b, c}, Integers] /. C[1] -> k]}];
a[n_] := a[n] = SelectFirst[Flatten[Table[r[n, k], {k, 1, kmax}], 1], GCD @@ # == 1 &] // First;
Table[Print[n, " ", a[n]]; a[n], {n, 1, 100}] (* Jean-François Alcover, Jan 01 2019 *)

cp's OEIS Frontend

A096033 Difference between leg and hypotenuse in primitive Pythagorean triangles.

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