A377017 -id:A377017 - cp's OEIS frontend

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.

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

Miguel-Ángel Pérez García-Ortega, Nov 05 2024

			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.

Miguel-Ángel Pérez García-Ortega, Capitulo 5. Catetos, El Libro de las Ternas Pitagóricas.

Cf. A002315, A377016, A377017, A377726, A385977 (long leg).

a(n) = 2*A002315(n) + 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

Miguel-Ángel Pérez García-Ortega, Nov 05 2024

			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.

Cf. A002315, A377016, A377017, A377725, A362545 (short legs).

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

a(n) = 2 * A002315(n) * (A002315(n) - 1).

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

Miguel-Ángel Pérez García-Ortega, Dec 12 2024

			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.

Cf. A002315, A377016, A377017, A377726, A378965.

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

a(n) = (A377726(n,1) + A377726(n,2) + A377726(n,3))/2.

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

Miguel-Ángel Pérez García-Ortega, Dec 12 2024

			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.

Cf. A002315, A377016, A377017, A377726, A378965.

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

a(n) = (A377726(n,1) * A377726(n,2))/2.

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

Miguel-Ángel Pérez García-Ortega, Dec 23 2024

			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.

Miguel-Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, preprint, 2024.

Cf. A002315, A377016, A377017, A377726, A378965, A378965.

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

a(n) = A377726(n,1) + A377726(n,2).

A380302 Area of the unique primitive Pythagorean triple whose inradius is the n-th odd prime and whose short leg is an even number.

Original entry on oeis.org

60, 210, 504, 1716, 2730, 5814, 7980, 13800, 26970, 32736, 54834, 74046, 85140, 110544, 157410, 215940, 238266, 314364, 373176, 405150, 511920, 592620, 728910, 941094, 1061106, 1124760, 1259604, 1330890, 1481430, 2097024, 2299836, 2627934, 2743860, 3374850, 3511656, 3944154

Offset: 0

Miguel-Ángel Pérez García-Ortega, Jan 19 2025

Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2025.

Miguel-Ángel Pérez García-Ortega, El Libro de las Ternas Pitagóricas

Cf. A367335, A380301, A377017, A378386, A378966.

a = Table[Prime[n], {n, 2, 42}]; Apply[Join, Map[{# (# + 1) (# + 2)} &, a]]

a(n) = ( A367335(n,1) * A367335(n,2) )/2.

cp's OEIS Frontend

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

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.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Examples

References

Crossrefs

Programs

Mathematica

Formula

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.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A380302 Area of the unique primitive Pythagorean triple whose inradius is the n-th odd prime and whose short leg is an even number.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Mathematica

Formula