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 1-2 of 2 results.

A157446 a(n) = 16*n^2 - n.

Original entry on oeis.org

15, 62, 141, 252, 395, 570, 777, 1016, 1287, 1590, 1925, 2292, 2691, 3122, 3585, 4080, 4607, 5166, 5757, 6380, 7035, 7722, 8441, 9192, 9975, 10790, 11637, 12516, 13427, 14370, 15345, 16352, 17391, 18462, 19565, 20700, 21867, 23066, 24297, 25560
Offset: 1

Views

Author

Vincenzo Librandi, Mar 01 2009

Keywords

Comments

The identity (2048*n^2 - 128*n + 1)^2 - (16*n^2 - n)*(512*n - 16)^2 = 1 can be written as A157448(n)^2 - a(n)*A157447(n)^2 = 1. - Vincenzo Librandi, Jan 26 2012
This is the case s=4 of the identity (8*n^2*s^4 - 8*n*s^2 + 1)^2 - (n^2*s^2 - n)*(8*n*s^3 - 4*s)^2 = 1. - Bruno Berselli, Jan 26 2012
Sequence found by reading the line from 15, in the direction 15, 62, ... in the square spiral whose vertices are the generalized decagonal numbers A074377. - Omar E. Pol, Nov 02 2012
The continued fraction expansion of sqrt(a(n)) is [4n-1; {1, 6, 1, 8n-2}]. For n=1, this collapses to [3; {1, 6}]. - Magus K. Chu, Sep 22 2022

Links

Crossrefs

Cf. A157447, A157448.

Programs

  • Magma
    I:=[15, 62, 141]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jan 26 2012
  • Mathematica
    LinearRecurrence[{3,-3,1},{15,62,141},40] (* Vincenzo Librandi, Jan 26 2012 *)
  • PARI
    for(n=1, 22, print1(16*n^2 - n", ")); \\ Vincenzo Librandi, Jan 26 2012

Formula

G.f.: x*(15 + 17*x)/(1-x)^3. - Vincenzo Librandi, Jan 26 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jan 26 2012

A157448 a(n) = 2048*n^2 - 128*n + 1.

Original entry on oeis.org

1921, 7937, 18049, 32257, 50561, 72961, 99457, 130049, 164737, 203521, 246401, 293377, 344449, 399617, 458881, 522241, 589697, 661249, 736897, 816641, 900481, 988417, 1080449, 1176577, 1276801, 1381121, 1489537, 1602049, 1718657, 1839361
Offset: 1

Views

Author

Vincenzo Librandi, Mar 01 2009

Keywords

Comments

The identity (2048*n^2 - 128*n + 1)^2 - (16*n^2 - n)*(512*n - 16)^2 = 1 can be written as a(n)^2 - A157446(n)*A157447(n)^2 = 1 (see also second comment at A157446). - Vincenzo Librandi, Jan 26 2012

Links

Crossrefs

Cf. A157446, A157447.

Programs

  • Magma
    I:=[1921, 7937, 18049]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..40]]; // Vincenzo Librandi, Jan 26 2012
  • Mathematica
    LinearRecurrence[{3,-3,1},{1921,7937,18049},40] (* Vincenzo Librandi, Jan 26 2012 *)
  • PARI
    for(n=1, 22, print1(2048*n^2 - 128*n + 1", ")); \\ Vincenzo Librandi, Jan 26 2012

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jan 26 2012
G.f.: x*(-1921 - 2174*x - x^2)/(x-1)^3. - Vincenzo Librandi, Jan 26 2012
Showing 1-2 of 2 results.

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