A377016 Semiperimeter of the unique primitive Pythagorean triple whose short leg is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

1, 28, 861, 28680, 970921, 32963140, 1119662181, 38034888528, 1292062686481, 43892073946540, 1491038320325421, 50651410052600280, 1720656899012149561, 58451683130389395028, 1985636569382856677301, 67453191675004485098400, 2291422880375627492063521, 77840924741066359629967420

Offset: 0

Author

Miguel-Ángel Pérez García-Ortega, Oct 13 2024

Keywords

• nonn
• easy

Comments

a(0) = 1 is included by convention. This corresponds to the Pythagorean triple 1^2 + 0^2 = 1^2.

Examples

			For n=2, the short leg is A002315(2) = 41 and the hypotenuse is A008844(n) = 841 so the semiperimeter is then a(2) = (41 + 840 + 841)/2 = 861.
		

References

• Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.

Links

• Index entries for linear recurrences with constant coefficients, signature (41,-246,246,-41,1).

Crossrefs

Cf. A002315, A078522, A008844.

Programs

Mathematica

s[n_]:=s[n]=Module[{a, b},a=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2;b=(a^2-1)/2;{(a+2b+1)/2}];semis={};Do[semis=Join[semis,FullSimplify[s[n]]],{n,0,17}];semis

PARI

Vec((1 - 13*x - 41*x^2 + 21*x^3)/((1 - 34*x + x^2)*(1 - 6*x + x^2)*(1 - x)) + O(x^20)) \\ Andrew Howroyd, Oct 14 2024

Formula

a(n) = (A002315(n) + 2*A008844(n) - 1)/2.

G.f.: (1 - 13*x - 41*x^2 + 21*x^3)/((1 - 34*x + x^2)*(1 - 6*x + x^2)*(1 - x)). - Andrew Howroyd, Oct 14 2024