cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Views

Author

Miguel-Ángel Pérez García-Ortega, Nov 05 2024

Keywords

Examples

			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.

Links

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

Crossrefs

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

Formula

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

A377017 Area of the unique primitive Pythagorean triple whose short leg is A002315(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

0, 84, 17220, 3412920, 675761016, 133797385260, 26491207202460, 5245125232676784, 1038508304885968560, 205619399242324129860, 40711602541676078766516, 8060691683852625858745320, 1595976241800278270688414120, 315995235184771245126273789084, 62565460590342906257639745449100
Offset: 0

Views

Author

Miguel-Ángel Pérez García-Ortega, Oct 13 2024

Keywords

Comments

a(0)=0 is included by convention. This corresponds to the Pythagorean triple 1^2 + 0^2 = 1^2.
All terms in this sequence are divisible by 84.

Examples

			For n=2, the short leg is A002315(2) = 41 and the long leg is A008844(2)-1 = 840 so the area is then a(2) = 41*840/2 = 17220.

References

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

Links

Crossrefs

Cf. A377016, A002315, A078522, A008844.

Programs

  • Mathematica
    s[n_]:=s[n]=Module[{a, b}, a=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2; b=(a^2-1)/2; {(a*b)/2}]; areas={}; Do[areas=Join[areas, FullSimplify[s[n]]], {n, 0, 17}]; areas

Formula

a(n) = A002315(n)*(A008844(n)-1)/2.
G.f.: 84*x*(1 + x)/((1 - 198*x + x^2)*(1 - 6*x + x^2)). - Andrew Howroyd, Oct 14 2024

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

Views

Author

Miguel-Ángel Pérez García-Ortega, Nov 05 2024

Keywords

Examples

			Triangles begins:
  n=1:     13,        84,        85;
  n=2:     81,      3280,      3281;
  n=3:    477,    113764,    113765;
  ...
This sequence gives the middle column.

References

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

Crossrefs

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

Programs

  • Mathematica
    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

Formula

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

Views

Author

Miguel-Ángel Pérez García-Ortega, Dec 12 2024

Keywords

Examples

			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.

References

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

Crossrefs

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

Programs

  • Mathematica
    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

Formula

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

A380301 Semiperimeter 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

20, 42, 72, 156, 210, 342, 420, 600, 930, 1056, 1482, 1806, 1980, 2352, 2970, 3660, 3906, 4692, 5256, 5550, 6480, 7140, 8190, 9702, 10506, 10920, 11772, 12210, 13110, 16512, 17556, 19182, 19740, 22650, 23256, 25122, 27060, 28392, 30450, 32580, 33306, 37056, 37830, 39402
Offset: 0

Views

Author

Miguel-Ángel Pérez García-Ortega, Jan 19 2025

Keywords

References

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

Links

Crossrefs

Cf. A367335, A377016, A378380, A378965, A379506.

Programs

  • Mathematica
    a = Table[Prime[n], {n, 2, 54}]; Apply[Join, Map[{#^2 + 3 # + 2} &, a]]

Formula

a(n) = ( A367335(n,1) + A367335(n,2) + A367335(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

Views

Author

Miguel-Ángel Pérez García-Ortega, Dec 12 2024

Keywords

Examples

			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.

References

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

Crossrefs

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

Programs

  • Mathematica
    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

Formula

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

Views

Author

Miguel-Ángel Pérez García-Ortega, Dec 23 2024

Keywords

Examples

			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.

Links

Crossrefs

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

Programs

  • Mathematica
    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

Formula

a(n) = A377726(n,1) + A377726(n,2).
Showing 1-7 of 7 results.

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