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 1-6 of 6 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

A001621 a(n) = n*(n + 1)*(n^2 + n + 2)/4.

Original entry on oeis.org

0, 2, 12, 42, 110, 240, 462, 812, 1332, 2070, 3080, 4422, 6162, 8372, 11130, 14520, 18632, 23562, 29412, 36290, 44310, 53592, 64262, 76452, 90300, 105950, 123552, 143262, 165242, 189660, 216690, 246512, 279312, 315282, 354620, 397530, 444222, 494912, 549822
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of integer sequences of length n+1 with sum zero and sum of absolute values 4. - R. H. Hardin, Feb 22 2009
Partial sums of A034262. - Jeremy Gardiner, Jun 23 2013

Links

Crossrefs

Cf. A002817, A058919, A092365.
Cf. A000124, A000217, A034262.

Programs

  • Mathematica
    a[n_]:=Sum[i+i^3, {i, 1, n}]; (* Vladimir Joseph Stephan Orlovsky, Dec 05 2008 *)
Array[# (# + 1) (#^2 + # + 2)/4 &, 39, 0] (* or *)
CoefficientList[Series[-2x (x^2 + x + 1)/(x - 1)^5, {x, 0, 38}], x] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {0, 2, 12, 42, 110}, 39] (* Robert G. Wilson v, Aug 05 2018 *)

Formula

Equals 2 * A002817 and (A058919(n-1) - 1)/2.
G.f.: (-2*x*(x^2+x+1))/(x-1)^5. - Maksym Voznyy (voznyy(AT)mail.ru), Jul 27 2009
a(n) = A000217(n) * A000124(n). - Torlach Rush, Aug 05 2018
E.g.f.: exp(x)*x*(8 + 16*x + 8*x^2 + x^3)/4. - Stefano Spezia, Oct 08 2022

A336535 a(n) = (m(n)^2 + 3)*(m(n)^2 + 7)/32, where m(n) = 2*n - 1.

Original entry on oeis.org

1, 6, 28, 91, 231, 496, 946, 1653, 2701, 4186, 6216, 8911, 12403, 16836, 22366, 29161, 37401, 47278, 58996, 72771, 88831, 107416, 128778, 153181, 180901, 212226, 247456, 286903, 330891, 379756, 433846, 493521, 559153, 631126, 709836, 795691, 889111, 990528, 1100386, 1219141, 1347261, 1485226, 1633528, 1792671
Offset: 1

Views

Author

Jeff Brown, Jul 24 2020

Keywords

Comments

For m(n) = 3,5,11, and 181, the perfect numbers (A000396), 6, 28, 496, and 33550336 are produced, respectively. 3,5,11, and 181 are the numbers m(n) such that (m(n)^2+7) is a power of 2. cf A038198.
The unique primitive Pythagorean triple whose inradius T(n) and its long leg and hypotenuse are consecutive natural numbers is (2*T(n)+1, 2*T(n)*(T(n)+1), 2*T(n)*(T(n)+1)+1) and its semiperimeter is (T(n)+1)*(2*T(n)+1) where T(n) = A000217(n). - Miguel-Ángel Pérez García-Ortega, May 16 2025

Examples

			m(2) = 2*2-1 = 3 and (3^2+3)*(3^2+7)/32 = 6, so 6 is in the sequence.

References

  • David M. Burton, Elementary Number Theory, McGraw-Hill (2011), 25.

Links

Crossrefs

Cf. A000396, A038198.
Cf. A002061, A014206.
Cf. A000217, A058919, A383833, A383834.

Programs

Formula

From Stefano Spezia, Jul 25 2020: (Start)
O.g.f.: x*(1 + x + 8*x^2 + x^3 + x^4)/(1 - x)^5.
a(n) = (1 - n + n^2)*(2 - n + n^2)/2.
a(n) = A002061(n)*A014206(n-1)/2.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5. (End)
a(n) = (A000217(n-1)+1)*(2*A000217(n-1)+1). - Miguel-Ángel Pérez García-Ortega, May 16 2025

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).

A131478 a(n) = ceiling(n^4/4).

Original entry on oeis.org

0, 1, 4, 21, 64, 157, 324, 601, 1024, 1641, 2500, 3661, 5184, 7141, 9604, 12657, 16384, 20881, 26244, 32581, 40000, 48621, 58564, 69961, 82944, 97657, 114244, 132861, 153664, 176821, 202500, 230881, 262144, 296481, 334084, 375157, 419904, 468541, 521284
Offset: 0

Views

Author

Mohammad K. Azarian, Jul 27 2007

Keywords

Links

Crossrefs

Cf. A000982, A004526, A007590, A008619, A036487, A058919.
Cf. A131479.

Programs

  • Magma
    [Ceiling(n^4/4) : n in [0..50]]; // Vincenzo Librandi, Oct 01 2011
  • Mathematica
    Ceiling[Range[0,40]^4/4] (* Harvey P. Dale, May 17 2019 *)
CoefficientList[Series[(x(x^3 + 6x^2 + 7x + 1)Cosh[x]+ (x^4 + 6x^3 + 7x^2 + x + 3)Sinh[x])/4,{x,0,35}],x]Table[n!,{n,0,35}] (* Stefano Spezia, Feb 19 2023 *)
  • PARI
    vector(50, n, n--;ceil(n^4/4)) \\ Michel Marcus, Jun 16 2015
  • Python
    def A131478(n): return n**4+3>>2 # Chai Wah Wu, Jan 30 2023

Formula

From R. J. Mathar, Dec 19 2008: (Start)
G.f.: x*(1 + 10*x^2 + x^4)/((1 - x)^5*(1 + x)).
a(n) + a(n+1) = A058919(n+1). (End)
a(n) = floor(n^4/4 + 3/4). - Bruno Berselli, Dec 21 2017
E.g.f.: (x*(x^3 + 6*x^2 + 7*x + 1)*cosh(x) + (x^4 + 6*x^3 + 7*x^2 + x + 3)*sinh(x))/4. - Stefano Spezia, Feb 18 2023

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 1-6 of 6 results.

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

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