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 1-3 of 3 results.

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

Views

Author

Miguel-Ángel Pérez García-Ortega, Nov 24 2024

Keywords

Examples

			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.

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, A377025, A378386

Programs

  • Mathematica
    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

Formula

a(n) = (A377725(n,1) + A377725(n,2) + A377725(n,3))/2.

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

Views

Author

Miguel-Ángel Pérez García-Ortega, Dec 23 2024

Keywords

Comments

For all n: a(n) == 7 (mod 8).

Examples

			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.

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. A002315, A377025, A378386, A378380.

Programs

  • Mathematica
    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

Formula

a(n) = A377725(n,1) + A377725(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

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, A380301, A377017, A378386, A378966.

Programs

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

Formula

a(n) = ( A367335(n,1) * A367335(n,2) )/2.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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