A385977
Length of the long leg of the triangles defined in A377725.
Original entry on oeis.org
4, 112, 3444, 114720, 3883684, 131852560, 4478648724, 152139554112, 5168250745924, 175568295786160, 5964153281301684, 202605640210401120, 6882627596048598244, 233806732521557580112, 7942546277531426709204, 269812766700017940393600, 9165691521502509968254084
Offset: 1
A377726
Lengths of the long leg 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.
Original entry on oeis.org
84, 3280, 113764, 3878112, 131820084, 4478459440, 152138450884, 5168244315840, 175568258308884, 5964153062868112, 202605638937276964, 6882627588628286880, 233806732478308836084, 7942546277279354556400, 269812766698548756220804, 9165691521493946935370112
Offset: 1
Triangles begins:
n=1: 13, 84, 85;
n=2: 81, 3280, 3281;
n=3: 477, 113764, 113765;
...
This sequence gives the middle column.
- 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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ra[n_]:=ra[n]=Module[{ra},ra=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2;{2ra-1,2ra^2-2ra,2ra^2-2ra+1}];exradio={};Do[exradio=Join[exradio,FullSimplify[ra[n]]],{n,0,10}];exradio
A378965
Semiperimeter of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.
Original entry on oeis.org
1, 91, 3321, 114003, 3879505, 131828203, 4478506761, 152138726691, 5168245923361, 175568267678203, 5964153117476505, 202605639255558003, 6882627590483364721, 233806732489121022091, 7942546277342372594601, 269812766698916052264003, 9165691521496087693591105, 311363698964228006760021403
Offset: 0
For n=2, the short leg is A377726(2,1) = 13, the long leg is A377725(2,2) = 842 and the hypotenuse is A377725(2,3) = 85 so the semiperimeter is then a(2) = (13 + 84 + 85)/2 = 91.
- Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.
-
s[n_]:=s[n]=Module[{ra},ra=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2;{ra(2ra-1)}];semis={};Do[semis=Join[semis,FullSimplify[s[n]]],{n,0,17}];semis
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.
-
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
A379508
Sum of the legs of the unique primitive Pythagorean triple (a,b,c) such that (a-b+c)/2 is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.
Original entry on oeis.org
1, 97, 3361, 114241, 3880897, 131836321, 4478554081, 152139002497, 5168247530881, 175568277047521, 5964153172084897, 202605639573839041, 6882627592338442561, 233806732499933208097, 7942546277405390632801, 269812766699283348307201, 9165691521498228451812097, 311363698964240484013304161
Offset: 0
For n=2, the short leg is A377726(2,1) = 13 and the long leg is A377725(2,2) = 84 so the semiperimeter is then a(2) = 13 + 84 = 97.
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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
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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