A377726 Length of the long leg of the unique primitive Pythagorean triple (x,y,z) such that (x-y+z)/2 is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

0, 84, 3280, 113764, 3878112, 131820084, 4478459440, 152138450884, 5168244315840, 175568258308884, 5964153062868112, 202605638937276964, 6882627588628286880, 233806732478308836084, 7942546277279354556400, 269812766698548756220804, 9165691521493946935370112

Offset

0,2

References

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Index entries for linear recurrences with constant coefficients, signature (41,-246,246,-41,1).

Formula

a(n) = 2 * A002315(n) * (A002315(n) - 1).

G.f.: 4*x*(3 + x)*(7 - 16*x + x^2)/((1 - x)*(1 - 34*x + x^2)*(1 - 6*x + x^2)). - Andrew Howroyd, Nov 17 2025

Example

Triangles begins:
  n=0:      1,         0,         1;
  n=1:     13,        84,        85;
  n=2:     81,      3280,      3281;
  n=3:    477,    113764,    113765;
  ...
This sequence gives the middle column.

Mathematica

ra[n_]:=ra[n]=Module[{ra}, ra=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2; {2ra-1, 2ra^2-2ra, 2ra^2-2ra+1}]; exradio={}; Do[exradio=Join[exradio, FullSimplify[ra[n]]], {n, 0, 10}]; exradio

Prog

(PARI) a(n)=my(t=polcoef((1 + x)/(1 - 6*x + x^2) + O(x*x^n), n)); 2*t*(t - 1); \\ Andrew Howroyd, Nov 17 2025

Crossrefs

Cf. A002315, A362545 (short leg), A377016, A377725, A378965 (semiperimeter), A378966 (area), A379508 (sum of legs).

Sequence in context: A219372 A026809 A114253 * A017800 A035737 A035806

Adjacent sequences: A377723 A377724 A377725 * A377727 A377728 A377729

Keyword

nonn,easy

Author

Miguel-Ángel Pérez García-Ortega, Nov 05 2024

Extensions

a(0)=0 prepended by Andrew Howroyd, Nov 17 2025

Status

approved