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 1-4 of 4 results.

A010014 a(0) = 1, a(n) = 24*n^2 + 2 for n>0.

Original entry on oeis.org

1, 26, 98, 218, 386, 602, 866, 1178, 1538, 1946, 2402, 2906, 3458, 4058, 4706, 5402, 6146, 6938, 7778, 8666, 9602, 10586, 11618, 12698, 13826, 15002, 16226, 17498, 18818, 20186, 21602, 23066, 24578, 26138, 27746, 29402, 31106, 32858, 34658, 36506, 38402, 40346
Offset: 0

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Author

N. J. A. Sloane

Keywords

Comments

Number of points of L_infinity norm n in the simple cubic lattice Z^3. - N. J. A. Sloane, Apr 15 2008
Numbers of cubes needed to completely "cover" another cube. - Xavier Acloque, Oct 20 2003
First bisection of A005897. After 1, all terms are in A000408. - Bruno Berselli, Feb 06 2012

Links

Crossrefs

Cf. A206399.

Programs

  • Mathematica
    Join[{1}, 24 Range[41]^2 + 2] (* Bruno Berselli, Feb 06 2012 *)
  • PARI
    a(n) = if (n==0, 1, 24*n^2 + 2);
vector(40, n, a(n-1)) \\ Altug Alkan, Sep 29 2015

Formula

a(n) = (2*n+1)^3 - (2*n-1)^3 for n >= 1. - Xavier Acloque, Oct 20 2003
G.f.: (1+x)*(1+22*x+x^2)/(1-x)^3. - Bruno Berselli, Feb 06 2012
a(n) = (2*n-1)^2 + (2*n+1)^2 + (4*n)^2 for n>0. - Bruno Berselli, Feb 06 2012
E.g.f.: (x*(x+1)*24+2)*exp(x)-1. - Gopinath A. R., Feb 14 2012
a(n) = A005899(n) + A195322(n), n > 0. - R. J. Cano, Sep 29 2015
Sum_{n>=0} 1/a(n) = 3/4 + sqrt(3)/24*Pi*coth(Pi*sqrt(3)/6) = 1.065052868574... - R. J. Mathar, May 07 2024
a(n) = 2*A158480(n), n>0. - R. J. Mathar, May 07 2024
a(n) = A069190(n)+A069190(n+1). - R. J. Mathar, May 07 2024

Extensions

More terms from Xavier Acloque, Oct 20 2003

A158479 a(n) = 36*n^2 + 6.

Original entry on oeis.org

42, 150, 330, 582, 906, 1302, 1770, 2310, 2922, 3606, 4362, 5190, 6090, 7062, 8106, 9222, 10410, 11670, 13002, 14406, 15882, 17430, 19050, 20742, 22506, 24342, 26250, 28230, 30282, 32406, 34602, 36870, 39210, 41622, 44106, 46662, 49290, 51990, 54762, 57606, 60522
Offset: 1

Views

Author

Vincenzo Librandi, Mar 20 2009

Keywords

Comments

The identity (12*n^2 + 1)^2 - (36*n^2 + 6)*(2*n)^2 = 1 can be written as A158480(n)^2 - a(n)*A005843(n)^2 = 1.

Links

Crossrefs

Cf. A005843, A158480, A227776.

Programs

  • Magma
    I:=[42, 150, 330]; [n le 3 select I[n] else 3*Self(n-1)-3*Self(n-2)+Self(n-3): n in [1..50]];
  • Mathematica
    LinearRecurrence[{3,-3,1},{42,150,330},40]
  • PARI
    a(n)=36*n^2+6 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f: 6*x*(7+4*x+x^2)/(1-x)^3.
From Amiram Eldar, Mar 05 2023: (Start)
Sum_{n>=1} 1/a(n) = (coth(Pi/sqrt(6))*Pi/sqrt(6) - 1)/12.
Sum_{n>=1} (-1)^(n+1)/a(n) = (1 - cosech(Pi/sqrt(6))*Pi/sqrt(6))/12. (End)
From Elmo R. Oliveira, Jan 15 2025: (Start)
E.g.f.: 6*(exp(x)*(6*x^2 + 6*x + 1) - 1).
a(n) = 6*A227776(n). (End)

A158546 a(n) = 144*n^2 + 12.

Original entry on oeis.org

12, 156, 588, 1308, 2316, 3612, 5196, 7068, 9228, 11676, 14412, 17436, 20748, 24348, 28236, 32412, 36876, 41628, 46668, 51996, 57612, 63516, 69708, 76188, 82956, 90012, 97356, 104988, 112908, 121116, 129612, 138396, 147468, 156828, 166476, 176412, 186636, 197148
Offset: 0

Views

Author

Vincenzo Librandi, Mar 21 2009

Keywords

Comments

The identity (24*n^2 + 1)^2 - (144*n^2 + 12)*(2*n)^2 = 1 can be written as A158547(n)^2 - a(n)*A005843(n)^2 = 1.

Links

Crossrefs

Cf. A005843, A158480, A158547.

Programs

  • Magma
    I:=[12, 156, 588]; [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, Feb 14 2012
  • Mathematica
    LinearRecurrence[{3, -3, 1}, {12, 156, 588}, 50] (* Vincenzo Librandi, Feb 14 2012 *)
144*Range[0,40]^2+12 (* Harvey P. Dale, Jul 14 2025 *)
  • PARI
    for(n=0, 40, print1(144*n^2 + 12", ")); \\ Vincenzo Librandi, Feb 14 2012

Formula

G.f.: 12*(1 + 10*x + 13*x^2)/(1-x)^3.
a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
From Amiram Eldar, Mar 06 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)) + 1)/24.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(2*sqrt(3)))*Pi/(2*sqrt(3)) + 1)/24. (End)
From Elmo R. Oliveira, Jan 15 2025: (Start)
E.g.f.: 12*exp(x)*(1 + 12*x + 12*x^2).
a(n) = 12*A158480(n). (End)

Extensions

Comment rewritten, a(0) added by R. J. Mathar, Oct 16 2009

A158638 a(n) = 48*n^2 + 1.

Original entry on oeis.org

1, 49, 193, 433, 769, 1201, 1729, 2353, 3073, 3889, 4801, 5809, 6913, 8113, 9409, 10801, 12289, 13873, 15553, 17329, 19201, 21169, 23233, 25393, 27649, 30001, 32449, 34993, 37633, 40369, 43201, 46129, 49153, 52273, 55489, 58801, 62209, 65713, 69313, 73009, 76801
Offset: 0

Views

Author

Vincenzo Librandi, Mar 23 2009

Keywords

Comments

The identity (48*n^2 + 1)^2 - (576*n^2 + 24)*(2*n)^2 = 1 can be written as a(n)^2 - A158637(n)*A005843(n)^2 = 1.

Links

Crossrefs

Cf. A000290, A005843, A158480, A158637.

Programs

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).
G.f.: -(1 + 46*x + 49*x^2)/(x-1)^3.
a(n) = 48*A000290(n) + 1. - Wesley Ivan Hurt, Dec 06 2013
From Amiram Eldar, Mar 19 2023: (Start)
Sum_{n>=0} 1/a(n) = (coth(Pi/(4*sqrt(3)))*Pi/(4*sqrt(3)) + 1)/2.
Sum_{n>=0} (-1)^n/a(n) = (cosech(Pi/(4*sqrt(3)))*Pi/(4*sqrt(3)) + 1)/2. (End)
From Elmo R. Oliveira, Jan 17 2025: (Start)
E.g.f.: exp(x)*(1 + 48*x + 48*x^2).
a(n) = A158480(2*n). (End)

Extensions

Comment rephrased and redundant formula replaced by R. J. Mathar, Oct 19 2009
Showing 1-4 of 4 results.

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

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