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URL: https://oeis.org/A379508

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A379508
Sum of the legs of the unique primitive Pythagorean triple (x,y,z) such that (x-y+z)/2 is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.
4
1, 97, 3361, 114241, 3880897, 131836321, 4478554081, 152139002497, 5168247530881, 175568277047521, 5964153172084897, 202605639573839041, 6882627592338442561, 233806732499933208097, 7942546277405390632801, 269812766699283348307201, 9165691521498228451812097, 311363698964240484013304161
OFFSET
0,2
LINKS
Miguel-Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, preprint, 2024.
FORMULA
a(n) = A362545(n) + A377726(n).
a(n) = 2*A002315(n)^2 - 1.
G.f.: (1 + 62*x + x^2)/((1 - x)*(1 - 34*x + x^2)). - Andrew Howroyd, Nov 17 2025
EXAMPLE
For n=1, the short leg is A362545(1) = 13 and the long leg is A377726(1) = 84 so the sum of the legs is then a(1) = 13 + 84 = 97.
MATHEMATICA
s[n_]:=s[n]=Module[{ra}, ra=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2; {2ra^2-1}]; sumas={}; Do[sumas=Join[semis, FullSimplify[s[n]]], {n, 0, 17}]; sumas
PROG
(PARI) a(n)=my(t=polcoef((1 + x)/(1 - 6*x + x^2) + O(x*x^n), n)); 2*t^2 - 1; \\ Andrew Howroyd, Nov 17 2025
CROSSREFS
Cf. A002315, A362545 (short leg), A377726 (long leg), A378965, A378966.
Sequence in context: A359639 A075665 A012839 * A017813 A017760 A069419
KEYWORD
nonn,easy
STATUS
approved