A382610 Area of the unique primitive Pythagorean triple whose inradius is A000045(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

0, 6, 6, 30, 84, 330, 1224, 4914, 19866, 82110, 341880, 1433790, 6034320, 25461774, 107592030, 455078910, 1925933100, 8153659170, 34527059160, 146226569946, 619340796690, 2623347596766, 11112097049136, 47070075918390, 199388054716704, 844610917608150, 3577801938273654, 15155740689781854, 64200560537978436

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 2025.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..1000

Index entries for linear recurrences with constant coefficients, signature (6,-2,-29,16,40,-11,-14,2,1).

Formula

a(n) = (A382608(n,1) * A382608(n,2))/2.

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

G.f.: 6*x*(1 - x)*(1 - 4*x - 3*x^2 + 12*x^3 + 6*x^4 - 3*x^5 - x^6)/((1 + x)*(1 - 3*x + x^2)*(1 + x - x^2)*(1 - x - x^2)*(1 - 4*x - x^2)). - Andrew Howroyd, Nov 13 2025

Example

For n=2, the short leg is A382608(2,1) = 3 and the long leg is A382608(2,2) = 4 so the area is then a(2) = (3 * 4 )/2 = 6.

Mathematica

a=Table[Fibonacci[n], {n, 0, 28}]; Apply[Join, Map[{#(#+1)(2#+1)}&, a]]

Prog

(PARI) a(n) = fibonacci(n)*(fibonacci(n) + 1)*(2*fibonacci(n) + 1); \\ Andrew Howroyd, Nov 13 2025

Crossrefs

Cf. A000045, A382608, A382609.

Sequence in context: A321528 A074002 A359741 * A381483 A140959 A015699

Adjacent sequences: A382607 A382608 A382609 * A382611 A382612 A382613

Keyword

nonn,easy

Author

Miguel-Ángel Pérez García-Ortega, Mar 31 2025

Status

approved