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 1-4 of 4 results.

A383834 Sum of the legs of the unique primitive Pythagorean triple whose inradius is A000217(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

1, 7, 31, 97, 241, 511, 967, 1681, 2737, 4231, 6271, 8977, 12481, 16927, 22471, 29281, 37537, 47431, 59167, 72961, 89041, 107647, 129031, 153457, 181201, 212551, 247807, 287281, 331297, 380191, 434311, 494017, 559681, 631687, 710431, 796321, 889777, 991231, 1101127, 1219921, 1348081
Offset: 0

Views

Author

Miguel-Ángel Pérez García-Ortega, May 11 2025

Keywords

Examples

			For n=1, the short leg is A002061(1) = 3 and the long leg is A212135(2) = 4 so the sum of the legs is then a(1) = 3 + 4 = 7.

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. A000217, A383833, A002061, A006007, A058919, A336535.

Programs

  • Mathematica
    a=Table[(n(n+1))/2,{n,0,40}];Apply[Join,Map[{2#^2+4#+1}&,a]]

Formula

a(n) = 2*(A000217(n))^2 + 4*A000217(n) + 1.
a(n) = 6*A006007(n) + 1

A383833 Area of the unique primitive Pythagorean triple whose inradius is A000217(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

0, 6, 84, 546, 2310, 7440, 19866, 46284, 97236, 188370, 341880, 588126, 967434, 1532076, 2348430, 3499320, 5086536, 7233534, 10088316, 13826490, 18654510, 24813096, 32580834, 42277956, 54270300, 68973450, 86857056, 108449334, 134341746, 165193860, 201738390
Offset: 0

Views

Author

Miguel-Ángel Pérez García-Ortega, May 11 2025

Keywords

Examples

			For n=1, the short leg is A002061(1) = 3 and the long leg is A212135(2) = 4 so the area is then a(1) = (3 * 4 )/2 = 6.

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. A000217, A002061, A058919, A383834, A336535.

Programs

  • Mathematica
    a=Table[(n(n+1))/2,{n,0,30}];Apply[Join,Map[{#(#+1)(2#+1)}&,a]]

Formula

a(n) = A000217(n) * (A000217(n) + 1) * (2*A000217(n) + 1).

A349919 Number of transitive relations on an n-set with exactly two ordered pairs.

Original entry on oeis.org

0, 0, 5, 27, 90, 230, 495, 945, 1652, 2700, 4185, 6215, 8910, 12402, 16835, 22365, 29160, 37400, 47277, 58995, 72770, 88830, 107415, 128777, 153180, 180900, 212225, 247455, 286902, 330890, 379755, 433845, 493520, 559152, 631125, 709835, 795690, 889110, 990527, 1100385, 1219140, 1347260, 1485225, 1633527, 1792670
Offset: 0

Views

Author

Firdous Ahmad Mala, Dec 05 2021

Keywords

Examples

			a(2) = 5. The five relations on a 2-set are {(1,1),(1,2)}, {(1,1),(2,1)}, {(1,1),(2,2)}, {(1,2),(2,2)} and {(2,1),(2,2)}.

Links

Crossrefs

Cf. A006905, A336535, A349927.
This is a diagonal of the array A285192.

Programs

  • Mathematica
    LinearRecurrence[{5,-10,10,-5,1},{0,0,5,27,90},50] (* Harvey P. Dale, Oct 23 2022 *)

Formula

a(n) = 5*C(n,2) + 12*C(n,3) + 12*C(n,4).
a(n) = (1/2)*(n^4 - 2*n^3 + 4*n^2 - 3*n).
a(n) = A336535(n) - 1.
From Elmo R. Oliveira, Aug 26 2025: (Start)
G.f.: x^2*(5 + 2*x + 5*x^2)/(1 - x)^5.
E.g.f.: x^2*(5 + 4*x + x^2)*exp(x)/2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5). (End)

A383957 Sum of the legs of the unique primitive Pythagorean triple whose inradius is A000108(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

7, 7, 17, 71, 449, 3697, 35377, 369799, 4095521, 47297537, 564278417, 6911822737, 86538816337, 1103803791601, 14305269324961, 187980077927431, 2500329797088481, 33615543666867361, 456277457385934801, 6246438372527004961, 86175353802778434481, 1197196443885744428881, 16738118900659230353761
Offset: 0

Views

Author

Miguel-Ángel Pérez García-Ortega, May 16 2025

Keywords

Examples

			For n=1, the short leg is A383251(1,1) = 3 and the long leg is A383251(1,2) = 4 so the sum of the legs is then a(1) = 3 + 4 = 7.

Links

Crossrefs

Cf. A000108, A383251, A381483, A382114, A006007, A058919, A336535.

Programs

  • Mathematica
    a=Table[(2n)!/(n!(n+1)!),{n,0,23}];Apply[Join,Map[{2#^2+4#+1}&,a]]

Formula

a(n) = A383251(n,1) + A383251(n,2).
a(n) = 2*(A000108(n))^2 + 4*A000108(n) + 1.
Showing 1-4 of 4 results.

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