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.

A382608 Long leg of the unique primitive Pythagorean triple whose inradius is A000045(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

4, 4, 12, 24, 60, 144, 364, 924, 2380, 6160, 16020, 41760, 109044, 285012, 745420, 1950312, 5104012, 13359280, 34969884, 91543980, 239651724, 627394464, 1642504612, 4300075584, 11257651300, 29472763684, 77160454284, 202008299064, 528863957340, 1384582787280
Offset: 1

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Examples

			Triangle begins:
  n=1:      3,    4,    5;
  n=2:      3,    4,    5;
  n=3:      5,   12,   13;
where this sequence is the middle column.
		

Crossrefs

Cf. A000045 (inradius), A001588 (short leg), A382609 (semiperimeter), A382610 (area).

Programs

  • Mathematica
    a=Table[Fibonacci[n],{n,0,16}];Apply[Join,Map[{2#+1,2#^2+2#,2#^2+2#+1}&,a]]

Formula

a(n) = 2*F(n)*(F(n) + 1) where F(n) = A000045(n).

A382609 Semiperimeter of the unique primitive Pythagorean triple whose inradius is A000045(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

1, 6, 6, 15, 28, 66, 153, 378, 946, 2415, 6216, 16110, 41905, 109278, 285390, 746031, 1951300, 5105610, 13361865, 34974066, 91550746, 239662671, 627412176, 1642533270, 4300121953, 11257726326, 29472885078, 77160650703, 202008616876, 528864471570, 1384583619321
Offset: 0

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Author

Keywords

Examples

			For n=2, the short leg is A382608(2,1) = 3, the long leg is A382608(2,2) = 4 and the hypotenuse is A382608(2,3) = 5 so the semiperimeter is then a(2) = (3 + 4 + 5)/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.

Crossrefs

Programs

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

Formula

a(n) = (A382608(n,1) + A382608(n,2) + A382608(n,3))/2.
a(n) = (Fibonacci(n) + 1)*(2*Fibonacci(n) + 1).

A382308 Sum of the legs of the unique primitive Pythagorean triple whose inradius is A000045(n) and such that its long leg and its hypotenuse are consecutive natural numbers.

Original entry on oeis.org

1, 7, 7, 17, 31, 71, 161, 391, 967, 2449, 6271, 16199, 42049, 109511, 285767, 746641, 1952287, 5107207, 13364449, 34978247, 91557511, 239673617, 627429887, 1642561927, 4300168321, 11257801351, 29473006471, 77160847121, 202008934687, 528864985799, 1384584451361
Offset: 0

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Author

Keywords

Examples

			For n=2, the short leg is A382608(2,1) = 3 and the long leg is A382608(2,2) = 4 so the sum of the legs is then a(2) = 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.

Crossrefs

Programs

  • Mathematica
    a=Table[Fibonacci[n],{n,0,30}];Apply[Join,Map[{2#^2+4#+1}&,a]]

Formula

a(n) = A382608(n,1) + A382608(n,2).
a(n) = 2*(Fibonacci(n))^2+4*Fibonacci(n) + 1.
Showing 1-3 of 3 results.