A378380
Semiperimeter of the unique primitive Pythagorean triple whose inradius is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.
Original entry on oeis.org
6, 120, 3486, 114960, 3885078, 131860680, 4478696046, 152139829920, 5168252353446, 175568305155480, 5964153335910078, 202605640528682160, 6882627597903676086, 233806732532369766120, 7942546277594444747406, 269812766700385236436800, 9165691521504650726475078, 311363698964277915773152440
Offset: 0
For n=2, the short leg is A377725(2,1) = 15, the long leg is A377725(2,2) = 112 and the hypotenuse is A377725(2,3) = 113 so the semiperimeter is then a(2) = (15 + 112 + 113)/2 = 120.
- Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.
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s[n_]:=s[n]=Module[{r},r=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2;{(r+1)(2r+1)}];semis={};Do[semis=Join[semis,FullSimplify[s[n]]],{n,0,17}];semis
A378386
Area of the unique primitive Pythagorean triple whose inradius is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.
Original entry on oeis.org
6, 840, 142926, 27475440, 5411913654, 1070576860920, 211936375592766, 41961230070745440, 8308074191463867366, 1644955457291036718120, 325692829279638552084654, 64485533774729467185564240, 12767809944726167559580210326, 2527961881828880059792526682840, 500523684734657069477415103656606
Offset: 0
For n=2, the short leg is A377725(2,1) = 15 and the long leg is A377725(2,2) = 112 so the area is then a(2) = (15 * 112)/2 = 840.
- Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.
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d[n_]:=d[n]=Module[{r},r=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2;{r(r+1)(2r+1)}];areas={};Do[areas=Join[areas,FullSimplify[d[n]]],{n,0,17}];areas
A379509
Sum of the legs of the unique primitive Pythagorean triple whose inradius is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.
Original entry on oeis.org
7, 127, 3527, 115199, 3886471, 131868799, 4478743367, 152140105727, 5168253960967, 175568314524799, 5964153390518471, 202605640846963199, 6882627599758753927, 233806732543181952127, 7942546277657462785607, 269812766700752532479999, 9165691521506791484696071, 311363698964290393026435199
Offset: 0
For n=2, the short leg is A377725(2,1) = 15 the long leg is A377725(2,2) = 112 so the semiperimeter is then a(2) = 15 + 112 = 127.
- Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.
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s[n_]:=s[n]=Module[{r},r=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2;{2r^2+4r+1}];sumas={};Do[semis=Join[sumas,FullSimplify[s[n]]],{n,0,17}];sumas
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