A384566 Area of the unique primitive Pythagorean triple whose inradius is A002378(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

0, 30, 546, 3900, 17220, 56730, 153510, 360696, 762120, 1482390, 2698410, 4652340, 7665996, 12156690, 18654510, 27821040, 40469520, 57586446, 80354610, 110177580, 148705620, 197863050, 259877046, 337307880, 433080600, 550518150, 693375930, 865877796, 1072753500, 1319277570, 1611309630

Offset

0,2

Comments

a(n) is multiple of 6 for all n.

Links

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

José Miguel Blanco Casado and Miguel Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas.

Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Formula

a(n) = (A384288(n,1) * A384288(n,2))/2.

a(n) = A002378(n)*(A002378(n) + 1)*(2*A002378(n) + 1).

From Andrew Howroyd, Nov 12 2025: (Start)

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

G.f.: 6*x*(5 + 56*x + 118*x^2 + 56*x^3 + 5*x^4)/(1 - x)^7. (End)

Example

For n=1, the short leg is A384288(1,1) = 5 and the long leg is A384288(1,2) = 12 so the area is then a(1) = (5 * 12 )/2 = 30.

Mathematica

a=Table[(n(n+1)), {n, 0, 30}]; Apply[Join, Map[{#(#+1)(2#+1)}&, a]]

Crossrefs

Cf. A002378, A384288, A008514, A237516.

Sequence in context: A037961 A377974 A143399 * A293103 A075510 A285168

Adjacent sequences: A384563 A384564 A384565 * A384567 A384568 A384569

Keyword

nonn,easy

Author

Miguel-Ángel Pérez García-Ortega, Jun 03 2025

Status

approved