A097265 -id:A097265 - cp's OEIS frontend

A097266 Number of primitive Pythagorean quadruples with diagonal 2n+1.

0, 1, 0, 1, 2, 2, 1, 2, 2, 3, 4, 3, 2, 5, 3, 4, 7, 4, 4, 6, 5, 6, 6, 6, 7, 9, 6, 6, 11, 8, 7, 12, 5, 9, 12, 9, 9, 10, 12, 10, 14, 11, 7, 14, 11, 12, 16, 10, 12, 19, 12, 13, 16, 14, 13, 18, 14, 12, 18, 16, 17, 21, 12, 16, 23, 17, 20, 18, 17, 18, 24, 18, 13, 28, 18, 19, 25, 16, 19, 26, 24

Offset: 0

Ray Chandler, Aug 16 2004

nonn

There are no such quadruples with diagonal 2n. - Michael Somos, Nov 17 2018

Eric Weisstein's World of Mathematics, Pythagorean Quadruple.

Cf. A096907-A096909, A097263-A097265.

a[ n_] := With[ {w = 2 n + 1}, Sum[ Boole[x^2 + y^2 + z^2 == w^2 && 1 == GCD[x, y, z, w]], {z, w - 1}, {y, z}, {x, y}]]; (* Michael Somos, Nov 17 2018 *)

{a(n) = my(w = 2*n+1); sum(z=1, w-1, sum(y=1, z, sum(x=1, y,  x^2 + y^2 + z^2 == w^2 && 1 == gcd([x, y, z, w]))))}; /* Michael Somos, Nov 17 2018 */

A360946 Number of Pythagorean quadruples with inradius n.

Original entry on oeis.org

1, 3, 6, 10, 9, 19, 16, 25, 29, 27, 27, 56, 31, 51, 49, 61, 42, 91, 52, 71, 89, 86, 63, 142, 64, 95, 116, 132, 83, 153, 90, 144, 149, 133, 108, 238, 108, 162, 169, 171, 122, 284, 130, 219, 200, 196, 145, 340, 174, 201, 231, 239, 164, 364, 176, 314, 278, 256, 190, 399, 195, 281, 360, 330

Offset: 1

Miguel-Ángel Pérez García-Ortega, Feb 26 2023

nonn

A Pythagorean quadruple is a quadruple (a,b,c,d) of positive integers such that a^2 + b^2 + c^2 = d^2 with a <= b <= c. Its inradius is (a+b+c-d)/2, which is a positive integer.

For every positive integer n, there is at least one Pythagorean quadruple with inradius n.

			For n=1 the a(1)=1 solution is (1,2,2,3).
For n=2 the a(2)=3 solutions are (1,4,8,9), (2,3,6,7) and (2,4,4,6).
For n=3 the a(3)=6 solutions are (1,6,18,19), (2,5,14,15), (2,6,9,11), (3,4,12,13), (3,6,6,9) and (4,4,7,9).

J. M. Blanco Casado, J. M. Sánchez Muñoz, and M. A. Pérez García-Ortega, El Libro de las Ternas Pitagóricas, Preprint 2023.

Miguel-Ángel Pérez García-Ortega, Pythagorean Quadruples (in Spanish).
Wikipedia, Pythagorean quadruple.

Cf. A096907, A096908, A096909, A096910, A097263, A097264, A097265, A097266, A097267

Cf. A169580, A225206, A225207, A230543, A330893, A330894, A331365, A335654, A359741

n=50;
div={};suc={};A={};
Do[A=Join[A,{Range[1,(1+1/Sqrt[3])q]}],{q,1,n}];
Do[suc=Join[suc,{Length[div]}];div={};For [i=1,i<=Length[Extract[A,q]],i++,div=Join[div,Intersection[Divisors[q^2+(Extract[Extract[A,q],i]-q)^2],Range[2(Extract[Extract[A,q],i]-q),Sqrt[q^2+(Extract[Extract[A,q],i]-q)^2]]]]],{q,1,n}];suc=Rest[Join[suc,{Length[div]}]];matriz={{"q"," ","cuaternas"}};For[j=1,j<=n,j++,matriz=Join[matriz,{{j," ",Extract[suc,j]}}]];MatrixForm[Transpose[matriz]]

cp's OEIS Frontend

A097266 Number of primitive Pythagorean quadruples with diagonal 2n+1.

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