A382409 Semiperimeter of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 is A000032(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

6, 1, 15, 28, 91, 231, 630, 1653, 4371, 11476, 30135, 79003, 207046, 542361, 1420455, 3719628, 9739491, 25500511, 66764790, 174798253, 457637131, 1198124676, 3136755615, 8212172403, 21499810566, 56287338481, 147362333055, 385799868028, 1010037606571, 2644313494551, 6922903755510

Offset: 0

Author

Miguel-Ángel Pérez García-Ortega, Mar 24 2025

Keywords

• nonn
• easy

Examples

			For n=3, the short leg is A382379(2,1) = 5, the long leg is A382379(2,2) = 12 and the hypotenuse is A382379(2,3) = 13 so the semiperimeter is then a(3) = (5 + 12 + 13)/2 = 15.
		

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 (3,1,-5,-1,1).

Crossrefs

Cf. A000032, A382379, A382410.

Programs

Mathematica

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

Formula

a(n) = (A382379(n,1) + A382379(n,2) + A382379(n,3))/2.

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