A378386 Area of the unique primitive Pythagorean triple whose inradius is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

6, 840, 142926, 27475440, 5411913654, 1070576860920, 211936375592766, 41961230070745440, 8308074191463867366, 1644955457291036718120, 325692829279638552084654, 64485533774729467185564240, 12767809944726167559580210326, 2527961881828880059792526682840, 500523684734657069477415103656606

Offset

0,1

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 (239,-8365,48995,-48995,8365,-239,1).

Formula

a(n) = A377725(n) * A385977(n) / 2.

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

G.f.: 6*(1 - 99*x - 1274*x^2 + 8126*x^3 - 391*x^4 - 91*x^5)/((1 - x)*(1 - 198*x + x^2)*(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 and the long leg is A385977(1) = 112 so the area is then a(1) = (15 * 112)/2 = 840.

Mathematica

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

Prog

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

Crossrefs

Cf. A002315, A377225, A378380, A379509, A385977.

Sequence in context: A271637 A249126 A337162 * A281690 A201141 A078927

Adjacent sequences: A378383 A378384 A378385 * A378387 A378388 A378389

Keyword

nonn,easy

Author

Miguel-Ángel Pérez García-Ortega, Nov 24 2024

Status

approved