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.

A157010 a(n) = 1681*n^2 - 756*n + 85.

Original entry on oeis.org

1010, 5297, 12946, 23957, 38330, 56065, 77162, 101621, 129442, 160625, 195170, 233077, 274346, 318977, 366970, 418325, 473042, 531121, 592562, 657365, 725530, 797057, 871946, 950197, 1031810, 1116785, 1205122, 1296821, 1391882, 1490305
Offset: 1

Views

Author

Vincenzo Librandi, Feb 23 2009

Keywords

Comments

The identity (5651522*n^2 -2541672*n +285769)^2 - (1681*n^2 -756*n +85) * (137842*n -30996)^2 = 1 can be written as (A157106(n))^2 - (a(n))*(A157105(n))^2 = 1.

Links

Crossrefs

Cf. A157105, A157106.

Programs

  • GAP
    List([1..40], n-> 1681*n^2 - 756*n + 85); # G. C. Greubel, Feb 23 2019
  • Magma
    I:=[1010, 5297, 12946]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];
  • Maple
    A157010:=n->1681*n^2 - 756*n + 85; seq(A157010(n), n=1..30); # Wesley Ivan Hurt, Jan 24 2014
  • Mathematica
    LinearRecurrence[{3,-3,1},{1010,5297,12946},30]
Table[1681n^2-756n+85,{n,40}] (* Harvey P. Dale, Sep 25 2024 *)
  • PARI
    a(n) = 1681*n^2 - 756*n + 85
  • Sage
    [1681*n^2 - 756*n + 85 for n in (1..40)] # G. C. Greubel, Feb 23 2019

Formula

a(n) = 3*a(n-1) -3*a(n-2) +a(n-3).
G.f: x*(1010 + 2267*x + 85*x^2)/(1-x)^3.
E.g.f.: -85 + (85 + 925*x + 1681*x^2)*exp(x). - G. C. Greubel, Feb 23 2019

A157106 a(n) = 5651522*n^2 - 2541672*n + 285769.

Original entry on oeis.org

3395619, 17808513, 43524451, 80543433, 128865459, 188490529, 259418643, 341649801, 435184003, 540021249, 656161539, 783604873, 922351251, 1072400673, 1233753139, 1406408649, 1590367203, 1785628801, 1992193443, 2210061129
Offset: 1

Views

Author

Vincenzo Librandi, Feb 23 2009

Keywords

Comments

The identity (5651522*n^2-2541672*n+285769)^2-(1681*n^2-756*n+85)*(137842*n-30996)^2=1 can be written as a(n)^2-A157010(n)*A157105(n)^2=1.

Links

Crossrefs

Cf. A157010, A157105.

Programs

  • Magma
    I:=[3395619, 17808513, 43524451]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]];
  • Maple
    A157106:=n->5651522*n^2 - 2541672*n + 285769; seq(A157106(n), n=1..30); # Wesley Ivan Hurt, Jan 23 2014
  • Mathematica
    LinearRecurrence[{3,-3,1},{3395619,17808513,43524451},30]
  • PARI
    a(n) = 5651522*n^2 - 2541672*n + 285769

Formula

a(n) = 3*a(n-1) -3*a(n-2) +a(n-3).

A157116 Numbers k such that k^2 + 1 == 0 (mod 41^2).

Original entry on oeis.org

378, 1303, 2059, 2984, 3740, 4665, 5421, 6346, 7102, 8027, 8783, 9708, 10464, 11389, 12145, 13070, 13826, 14751, 15507, 16432, 17188, 18113, 18869, 19794, 20550, 21475, 22231, 23156, 23912, 24837, 25593, 26518, 27274, 28199, 28955, 29880
Offset: 1

Views

Author

Vincenzo Librandi, Feb 23 2009

Keywords

Examples

			378^2 + 1 == 0 (mod 41^2).
1303^2 + 1 == 0 (mod 41^2).
2059^2 + 1 == 0 (mod 41^2).

Links

Crossrefs

Cf. A157105, A157111.

Programs

  • Magma
    [(3362*n-1681+169*(-1)^n)/4: n in [1..40]]; // Vincenzo Librandi, Sep 11 2013
  • Mathematica
    CoefficientList[Series[(14 x + 27) (27 x + 14) / ((1 + x) (x - 1)^2), {x, 0, 40}], x] (* Vincenzo Librandi, Sep 11 2013 *)
Select[Range[30000],PowerMod[#,2,1681]==1680&] (* or *) LinearRecurrence[ {1,1,-1},{378,1303,2059},40] (* Harvey P. Dale, Jul 05 2021 *)

Formula

a(1)=378, a(2)=1303; a(n) = 2*a(n-1) - a(n-2) - 13^2 if n is odd, and a(n) = 2*a(n-1) - a(n-2) + 13^2 if n is even.
From R. J. Mathar, Mar 08 2009: (Start)
a(n) = (3362n - 1681 + 169*(-1)^n)/4.
G.f.: x*(14*x+27)*(27*x+14)/((1+x)*(x-1)^2). (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = cot(378*Pi/1681)*Pi/1681. - Amiram Eldar, Feb 26 2023
Showing 1-3 of 3 results.

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