A381721 Sum of the legs of the unique primitive Pythagorean triple whose inradius is A000032(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

17, 7, 31, 49, 127, 287, 721, 1799, 4607, 11857, 30751, 79999, 208657, 544967, 1424671, 3726449, 9750527, 25518367, 66793681, 174844999, 457712767, 1198247057, 3136953631, 8212492799, 21500328977, 56288177287, 147363690271, 385802064049, 1010041159807, 2644319243807, 6922913058001, 18124414244999, 47450320478207

Offset: 0

Author

Miguel-Ángel Pérez García-Ortega, Mar 05 2025

Keywords

• nonn
• easy

Examples

			For n=2, the short leg is A380821(2,1) = 7 and the long leg is A380821(2,2) = 24 so the semiperimeter is then a(2) = 7 + 24 = 31.
		

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, A380823, A380824, A000032.

Programs

Mathematica

a=Table[LucasL[n],{n,0,35}];Apply[Join,Map[{2#^2+4#+1}&,a]]

Formula

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

a(n) = 2*(Lucas(n))^2 + 4*Lucas(n) + 1.

G.f.: (x^5-33*x^4+41*x^3+37*x^2-61*x+17)/((x-1)*(x+1)*(x^2-3*x+1)*(x^2+x-1)). - Alois P. Heinz, Jun 16 2025

Extensions

a(19) corrected by Georg Fischer, Jun 16 2025