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.

A157440 a(n) = 121*n^2 - 204*n + 86.

Original entry on oeis.org

3, 162, 563, 1206, 2091, 3218, 4587, 6198, 8051, 10146, 12483, 15062, 17883, 20946, 24251, 27798, 31587, 35618, 39891, 44406, 49163, 54162, 59403, 64886, 70611, 76578, 82787, 89238, 95931, 102866, 110043, 117462, 125123, 133026, 141171
Offset: 1

Views

Author

Vincenzo Librandi, Mar 01 2009

Keywords

Comments

The identity (14641*n^2 - 24684*n + 10405)^2 - (121*n^2 - 204*n + 86)*(1331*n - 1122)^2 = 1 can be written as A157442(n)^2 - a(n)*A157441(n)^2 = 1. - Vincenzo Librandi, Jan 29 2012
The continued fraction expansion of sqrt(a(n)) is [11n-10; {1, 2, 1, 2, 11n-10, 2, 1, 2, 1, 22n-20}]. For n=1, this collapses to [1; {1, 2}]. - Magus K. Chu, Sep 13 2022

Links

Crossrefs

Cf. A157441, A157442.

Programs

  • Magma
    I:=[3, 162, 563]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Jan 29 2012
  • Mathematica
    LinearRecurrence[{3,-3,1},{3,162,563},50] (* Vincenzo Librandi, Jan 29 2012 *)
  • PARI
    a(n)=121*n^2-204*n+86 \\ Charles R Greathouse IV, Dec 28 2011

Formula

G.f.: x*(-3 - 153*x - 86*x^2)/(x-1)^3. - Vincenzo Librandi, Jan 29 2012
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - Vincenzo Librandi, Jan 29 2012

A157442 a(n) = 14641*n^2 - 24684*n + 10405.

Original entry on oeis.org

362, 19601, 68122, 145925, 253010, 389377, 555026, 749957, 974170, 1227665, 1510442, 1822501, 2163842, 2534465, 2934370, 3363557, 3822026, 4309777, 4826810, 5373125, 5948722, 6553601, 7187762, 7851205, 8543930, 9265937
Offset: 1

Views

Author

Vincenzo Librandi, Mar 01 2009

Keywords

Comments

The identity (14641*n^2 - 24684*n + 10405)^2 - (121*n^2 - 204*n + 86)*(1331*n - 1122)^2 = 1 can be written as a(n)^2 - A157440(n)*A157441(n)^2 = 1. - Vincenzo Librandi, Jan 29 2012

Links

Crossrefs

Cf. A017485, A157440, A157441.

Programs

  • Magma
    I:=[362, 19601, 68122]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+1*Self(n-3): n in [1..50]]; // Vincenzo Librandi, Jan 29 2012
  • Mathematica
    Table[14641n^2-24684n+10405,{n,30}] (* or *) LinearRecurrence[{3,-3,1},{362,19601,68122},30]
  • PARI
    for(n=1, 40, print1(14641*n^2 - 24684*n + 10405", ")); \\ Vincenzo Librandi, Jan 29 2012

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3); a(1)=362, a(2)=19601, a(3)=68122. - Harvey P. Dale, Oct 22 2011
G.f.: x*(-10405*x^2 - 18515*x - 362)/(x-1)^3. - Harvey P. Dale, Oct 22 2011
a(n) = A017485(11*n-10)^2 + 1. - Bruno Berselli, Jan 29 2012
Showing 1-2 of 2 results.

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

Content is available under The OEIS End-User License Agreement.