A379509 Sum of the legs of the unique primitive Pythagorean triple whose inradius is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

7, 127, 3527, 115199, 3886471, 131868799, 4478743367, 152140105727, 5168253960967, 175568314524799, 5964153390518471, 202605640846963199, 6882627599758753927, 233806732543181952127, 7942546277657462785607, 269812766700752532479999, 9165691521506791484696071, 311363698964290393026435199

Offset

0,1

Comments

For all n: a(n) == 7 (mod 8).

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

Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas.

Index entries for linear recurrences with constant coefficients, signature (40,-206,40,-1).

Formula

a(n) = A377725(n) + A385977(n).

a(n) = 2*A002315(n)^2 + 4*A002315(n) + 1.

G.f.: (7 - 153*x - 111*x^2 + x^3)/((1 - 34*x + x^2)*(1 - 6*x + x^2)). - Andrew Howroyd, Nov 16 2025

Example

For n=1, the short leg is A377725(1) = 15 the long leg is A385977(1) = 112 so the sum of the legs is then a(1) = 15 + 112 = 127.

Mathematica

s[n_]:=s[n]=Module[{r}, r=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2; {2r^2+4r+1}]; sumas={}; Do[semis=Join[sumas, FullSimplify[s[n]]], {n, 0, 17}]; sumas

Prog

(PARI) a(n)=my(t=polcoef((1 + x)/(1 - 6*x + x^2) + O(x*x^n), n)); 2*t^2 + 4*t + 1; \\ Andrew Howroyd, Nov 16 2025

Crossrefs

Cf. A002315, A377725, A378386, A378380, A385977.

Sequence in context: A139291 A274673 A363870 * A382059 A215066 A092676

Adjacent sequences: A379506 A379507 A379508 * A379510 A379511 A379512

Keyword

nonn,easy

Author

Miguel-Ángel Pérez García-Ortega, Dec 23 2024

Status

approved