OFFSET
1,2
COMMENTS
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.
REFERENCES
Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.
LINKS
Miguel-Ángel Pérez García-Ortega, Contando y calculando cuaternas pitagóricas
Index entries for linear recurrences with constant coefficients, signature (1,-1,1,2,-2,2,-2,-1,1,-1,1).
FORMULA
Row n = (a, b, c, d) = (n, n + 1, n^2 + n, n^2 + n + 1).
G.f.: -x*(x^10-x^9+2*x^8-2*x^7-3*x^4+2*x^3+x^2+x+1)/((x+1)^2*(x-1)^3*(x^2+1)^3). - Alois P. Heinz, Sep 18 2024
EXAMPLE
Table begins:
n=1: 1, 2, 2, 3;
n=2: 2, 3, 6, 7;
n=3: 3, 4, 12, 13;
n=4: 4, 5, 20, 21;
n=5: 6, 7, 42, 43;
MATHEMATICA
cuaternas={}; Do[cuaternas=Join[cuaternas, {n, n+1, n^2+n, n^2+n+1}], {n, 1, 35}]; cuaternas
Table[{n, n+1, n^2+n, n^2+n+1}, {n, 20}]//Flatten (* Harvey P. Dale, Sep 06 2024 *)
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Miguel-Ángel Pérez García-Ortega, Mar 27 2024
STATUS
approved