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.

A167191 a(n) = 4*n*(1 + 45*n + 620*n^2).

Original entry on oeis.org

2664, 20568, 68592, 161616, 314520, 542184, 859488, 1281312, 1822536, 2498040, 3322704, 4311408, 5479032, 6840456, 8410560, 10204224, 12236328, 14521752, 17075376, 19912080, 23046744, 26494248, 30269472, 34387296, 38862600, 43710264, 48945168, 54582192, 60636216
Offset: 1

Views

Author

A.K. Devaraj, Oct 30 2009

Keywords

Comments

See A167190, where this sequence arises as the integer part of the quotient.

Links

Crossrefs

Cf. A167190.

Programs

  • Magma
    I:=[2664, 20568, 68592, 161616]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..40]]; // Vincenzo Librandi, Jul 02 2012
  • Mathematica
    CoefficientList[Series[24*(111+413*x+96*x^2)/(x-1)^4,{x,0,40}],x] (* Vincenzo Librandi, Jul 02 2012 *)
LinearRecurrence[{4,-6,4,-1},{2664,20568,68592,161616},40] (* Harvey P. Dale, Jun 15 2014 *)

Formula

G.f.: 24*x*(111+413*x+96*x^2)/(x-1)^4. - R. J. Mathar, Jan 27 2012
a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). - Vincenzo Librandi, Jul 02 2012
E.g.f.: 4*x*(666 + 1905*x + 620*x^2)*exp(x). - Elmo R. Oliveira, Aug 07 2025

Extensions

Extended beyond a(6) by R. J. Mathar, Nov 17 2009

A167467 a(n) = 25*n^3 - n*(5*n+1)/2 + 1.

Original entry on oeis.org

23, 190, 652, 1559, 3061, 5308, 8450, 12637, 18019, 24746, 32968, 42835, 54497, 68104, 83806, 101753, 122095, 144982, 170564, 198991, 230413, 264980, 302842, 344149, 389051, 437698, 490240, 546827, 607609, 672736, 742358, 816625, 895687, 979694, 1068796
Offset: 1

Views

Author

A.K. Devaraj, Nov 05 2009

Keywords

Comments

Also the real part of f(x+n*f(x,y,z), y+n*f(x,y,z), z+n*f(x,y,z))/f(x,y,z) for f(x,y,z) = x^3+y^2+z at x=(-1+i*sqrt(3))/2, y=i and z=5.
If f(x,y,z) is a trivariate polynomial, f(x+n*f(x,y,z),y+n*f(x,y,z),z+n*f(x,y,z)) is congruent to 0 (mod f(x,y,z)).
The ratio f(x+n*f,y+n*f,z+n*f)/f of these two functions is decomposed into the real part (this sequence here), and the imaginary part. The imaginary part is 2*n*i + sqrt(3)*A167469(n)*i, where i=sqrt(-1) is the imaginary unit.

Examples

			f(x +f(x,y,z), y + f(x,y,z), z + f(x,y,z)) = (23 + 2i + 6*sqrt(3)*i)* f(x,y,z) at n=1.

Links

Crossrefs

Cf. A165806, A165808, A165809, A166715, A166957, A167190.

Programs

  • GAP
    List([1..50], n-> 25*n^3 - n*(5*n+1)/2 + 1); # G. C. Greubel, Sep 01 2019
  • Magma
    [25*n^3 - n*(5*n+1)/2 + 1: n in [1..50]]; // G. C. Greubel, Sep 01 2019
  • Maple
    f := proc(x,y,z) x^3+y^2+z ; end proc:
A167467 := proc(n) local rho,a ,x,y,z; a := f(x+n*f(x,y,z),y+n*f(x,y,z),z+n*f(x,y,z))/f(x,y,z) ; rho := (-1+I*sqrt(3))/2 ; a := subs({x = rho, y=I,z=5},a) ; a := expand(a) ; Re(a) ; end:
seq(A167467(n),n=1..50) ; # R. J. Mathar, Nov 12 2009
  • Mathematica
    LinearRecurrence[{4,-6,4,-1},{23,190,652,1559},50] (* Harvey P. Dale, Sep 28 2012 *)
  • PARI
    a(n)=1+25*n^3-n*(5*n+1)/2 \\ Charles R Greathouse IV, Jul 07 2013
  • Sage
    [25*n^3 - n*(5*n+1)/2 + 1 for n in (1..50)] # G. C. Greubel, Sep 01 2019

Formula

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
G.f.: x*(23 + 98*x + 30*x^2 - x^3)/(1-x)^4.
E.g.f.: (2 + 44*x + 145*x^2 + 50*x^3)*exp(x)/2 -1. - G. C. Greubel, Apr 09 2016

Extensions

a(2) and a(3) corrected, definition simplified and sequence extended by R. J. Mathar, Nov 12 2009
Showing 1-2 of 2 results.

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