A377725
Length of the short leg 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
3, 15, 83, 479, 2787, 16239, 94643, 551615, 3215043, 18738639, 109216787, 636562079, 3710155683, 21624372015, 126036076403, 734592086399, 4281516441987, 24954506565519, 145445522951123, 847718631141215, 4940866263896163, 28797478952235759, 167844007449518387
Offset: 1
Triangles begin:
n=1: 3, 4, 5;
n=2: 15, 112, 113;
n=3: 83, 3444, 3445;
n=4: 479, 114720, 114721;
...
This sequence gives the first column.
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
A378966
Area 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
0, 546, 132840, 27132714, 5400270960, 1070181351954, 211922939930520, 41960773653737946, 8308058686721274720, 1644954930586205575554, 325692811387179035829960, 64485533166912548464047114, 12767809924078284782564882640, 2527961881127459862292727058546, 500523684710829430645198931758200
Offset: 0
For n=2, the short leg is A377726(2,1) = 13 and the long leg so the semiperimeter is then a(2) = (13 * 84)/2 =546.
- Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.
-
ar[n_]:=ar[n]= Module[{ra},ra=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2;{ra(ra-1)(2ra-1)}];areas={};Do[areas=Join[areas,FullSimplify[ar[n]]],{n,0,16}];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
Showing 1-4 of 4 results.