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.

A156639 a(n) = 169*n^2 - 140*n + 29.

Original entry on oeis.org

58, 425, 1130, 2173, 3554, 5273, 7330, 9725, 12458, 15529, 18938, 22685, 26770, 31193, 35954, 41053, 46490, 52265, 58378, 64829, 71618, 78745, 86210, 94013, 102154, 110633, 119450, 128605, 138098, 147929, 158098
Offset: 1

Views

Author

Vincenzo Librandi, Feb 15 2009

Keywords

Comments

The identity (57122*n^2 - 47320*n + 9801)^2 - (169*n^2 - 140*n + 29)*(4394*n - 1820)^2 = 1 can be written as A156721(n)^2 - a(n)*A156627(n)^2 = 1.
The continued fraction expansion of sqrt(a(n)) is [13n-6; {1, 1, 1, 1, 1, 1, 26n-12}]. - Magus K. Chu, Sep 06 2022

Links

Crossrefs

Cf. A156627, A156721.

Programs

  • Magma
    I:=[58, 425, 1130]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..40]];
  • Mathematica
    LinearRecurrence[{3, -3, 1}, {58, 425, 1130}, 40]
Table[169n^2-140n+29,{n,40}] (* Harvey P. Dale, Mar 24 2023 *)
  • PARI
    a(n)=169*n^2-140*n+29 \\ Charles R Greathouse IV, Dec 23 2011

Formula

G.f.: x*(58 + 251*x + 29*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

Extensions

Edited by Charles R Greathouse IV, Jul 25 2010

A156718 Numbers k such that k^2 == -1 (mod 13^2).

Original entry on oeis.org

70, 99, 239, 268, 408, 437, 577, 606, 746, 775, 915, 944, 1084, 1113, 1253, 1282, 1422, 1451, 1591, 1620, 1760, 1789, 1929, 1958, 2098, 2127, 2267, 2296, 2436, 2465, 2605, 2634, 2774, 2803, 2943, 2972, 3112, 3141, 3281, 3310, 3450, 3479, 3619, 3648, 3788
Offset: 1

Views

Author

Vincenzo Librandi, Feb 14 2009

Keywords

Comments

Also, numbers of the form 169k +- 70.

Links

Crossrefs

Cf. A156636, A156640, A156627, A156639.

Programs

  • Magma
    [Floor(n/2)*169-70*(-1)^n: n in [1..50]];
  • Mathematica
    LinearRecurrence[{1,1,-1},{70,99,239},50]
  • PARI
    a(n)=n\2*169-70*(-1)^n \\ Charles R Greathouse IV, Dec 23 2011

Formula

a(n) = a(n-1) + a(n-2) - a(n-3).
G.f.: x*(70 + 29*x + 70*x^2) / ( (1+x)*(x-1)^2 ). - Alexander R. Povolotsky, Feb 15 2009
Sum_{n>=1} (-1)^(n+1)/a(n) = tan(29*Pi/338)*Pi/169. - Amiram Eldar, Feb 26 2023

A156721 a(n) = 57122*n^2 - 47320*n + 9801.

Original entry on oeis.org

19603, 143649, 381939, 734473, 1201251, 1782273, 2477539, 3287049, 4210803, 5248801, 6401043, 7667529, 9048259, 10543233, 12152451, 13875913, 15713619, 17665569, 19731763, 21912201, 24206883, 26615809, 29138979
Offset: 1

Views

Author

Vincenzo Librandi, Feb 15 2009

Keywords

Comments

The identity (57122*n^2 - 47320*n+9801)^2 - (169*n^2 - 140*n + 29)*(4394*n - 1820)^2 = 1 can be written as a(n)^2 - A156639(n)*A156627(n)^2 = 1.
This is the case s=13 and r=70 of the identity (2*(s^2*n-r)^2+1)^2 - (((s^2*n-r)^2+1)/s^2)*(2*s*(s^2*n-r))^2 = 1, where ((s^2*n-r)^2+1)/s^2 is an integer if r^2 == -1 (mod s^2). Therefore, for s=13, nonnegative r values are: 70, 99, 239, 268, 408, 437, 577, 606, 746, 775, 915, 944, ... - Bruno Berselli, Apr 24 2018

Links

Crossrefs

Cf. A156627, A156639.

Programs

  • Magma
    I:=[19603, 143649, 381939]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..40]];
  • Mathematica
    LinearRecurrence[{3, -3, 1}, {19603, 143649, 381939}, 40]
  • PARI
    a(n)=57122*n^2-47320*n+9801 \\ Charles R Greathouse IV, Dec 23 2011

Formula

G.f.: x*(19603 + 84840*x + 9801*x^2)/(1 - x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

Extensions

Edited by Charles R Greathouse IV, Jul 25 2010
Showing 1-3 of 3 results.

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