A383251
Short leg of the unique primitive Pythagorean triple whose inradius is A000108(n) and such that its long leg and its hypotenuse are consecutive natural numbers.
Original entry on oeis.org
3, 3, 5, 11, 29, 85, 265, 859, 2861, 9725, 33593, 117573, 416025, 1485801, 5348881, 19389691, 70715341, 259289581, 955277401, 3534526381, 13128240841, 48932534041, 182965127281, 686119227301, 2579808294649, 9723892802905, 36734706144305, 139067101832009
Offset: 0
Triangles begin:
n=0: 3, 4, 5;
n=1: 3, 4, 5;
n=2: 5, 12, 13;
n=3: 11, 60, 61;
...
This sequences gives the first column.
A381483
Area of the unique primitive Pythagorean triple whose inradius is A000108(n) and such that its long leg and its hypotenuse are consecutive natural numbers.
Original entry on oeis.org
6, 6, 30, 330, 6090, 153510, 4652340, 158459730, 5854550130, 229936985850, 9477338186316, 406314955623486, 18001068994899900, 820015284879972900, 38258577340819383240, 1822437624604345219170, 88405834606456644170370, 4358080082619077400555090, 217935771356984568896708700
Offset: 0
For n=2, the short leg is A382608(2,1) = 3 and the long leg is A382608(2,2) = 4 so the area is then a(2) = (3 * 4 )/2 = 6.
- Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.
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a=Table[(2n)!/(n!(n+1)!),{n,0,18}];Apply[Join,Map[{#(#+1)(2#+1)}&,a]]
A383957
Sum of the legs of the unique primitive Pythagorean triple whose inradius is A000108(n) and such that its long leg and its hypotenuse are consecutive natural numbers.
Original entry on oeis.org
7, 7, 17, 71, 449, 3697, 35377, 369799, 4095521, 47297537, 564278417, 6911822737, 86538816337, 1103803791601, 14305269324961, 187980077927431, 2500329797088481, 33615543666867361, 456277457385934801, 6246438372527004961, 86175353802778434481, 1197196443885744428881, 16738118900659230353761
Offset: 0
For n=1, the short leg is A383251(1,1) = 3 and the long leg is A383251(1,2) = 4 so the sum of the legs is then a(1) = 3 + 4 = 7.
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a=Table[(2n)!/(n!(n+1)!),{n,0,23}];Apply[Join,Map[{2#^2+4#+1}&,a]]
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