A380823 Semiperimeter of the unique primitive Pythagorean triple whose inradius is A000032(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

15, 6, 28, 45, 120, 276, 703, 1770, 4560, 11781, 30628, 79800, 208335, 544446, 1423828, 3725085, 9748320, 25514796, 66787903, 174835650, 457697640, 1198222581, 3136914028, 8212428720, 21500225295, 56288009526, 147363418828, 385801624845, 1010040449160, 2644318093956, 6922911197503

Offset: 0

Author

Miguel-Ángel Pérez García-Ortega, Feb 04 2025

Keywords

• nonn
• easy

Examples

			For n=2, the short leg is A380821(2,1) = 7, the long leg is A380821(2,2) = 24 and the hypotenuse is A380821(2,3) = 25 so the semiperimeter is then a(2) = (7 + 24 + 25)/2 = 28.
		

References

• Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.

Links

• Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas.
• Index entries for linear recurrences with constant coefficients, signature (4,-2,-6,4,2,-1).

Crossrefs

Cf. A380821, A380824, A381721, A000032.

Programs

Mathematica

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

Formula

a(n) = (A380821(n,1) + A380821(n,2) + A380821(n,3))/2.

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

G.f.: (15 - 54*x + 34*x^2 + 35*x^3 - 28*x^4)/((1 - x)*(1 + x)*(1 - 3*x + x^2)*(1 - x - x^2)). - Stefano Spezia, Mar 08 2025