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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.

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A167190 a(n) = 1 + 85*n + 2232*n^2 + 15276*n^3.

Original entry on oeis.org

17594, 131307, 432796, 1013717, 1965726, 3380479, 5349632, 7964841, 11317762, 15500051, 20603364, 26719357, 33939686, 42356007, 52059976, 63143249, 75697482, 89814331, 105585452, 123102501, 142457134, 163741007, 187045776
Offset: 1

Views

Author

A.K. Devaraj, Oct 30 2009

Keywords

Comments

As mentioned in A166957, polynomials in two variables, not necessarily homogeneous, also have a property similar to that in a single variable (cf. A165806, A165808 and A165809) viz f(x+k*f(x,y), y+k*f(x,y)) == 0 (mod f(x,y)). The quotient has two parts: a rational integer and a rational integer coefficient of sqrt(-1), when x belongs to Z(x = 5) and y is complex (sqrt(-1)). The polynomial considered is identical with that in A166957 viz x^3 + 2*x*y + y^2. The present sequence is only that of the rational integers and seq A167191 will consist of rational integer coefficients of sqrt(-1). Note: k belongs to N.

Examples

			When x = 5 and y = i, f(x,y) = x^3 + 2xy + y^2 = 124 + 10i. The quotient of f(x + f(x,y), y + f(x,y))/(124 + 10i) is 17594 + 2664i.

Links

Crossrefs

Cf. A165806, A165808, A165809, A166957.

Programs

  • GAP
    List([0..40], n-> 1 +85*n +2232*n^2 +15276*n^3); # G. C. Greubel, Sep 01 2019
  • Magma
    I:=[17594, 131307, 432796, 1013717]; [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
  • Maple
    seq(1 + 85*n + 2232*n^2 + 15276*n^3, n=1..40); # G. C. Greubel, Sep 01 2019
  • Mathematica
    CoefficientList[Series[(17594+60931*x+13132*x^2-x^3)/(x-1)^4,{x,0,40}],x] (* Vincenzo Librandi, Jul 02 2012 *)
Table[1 +85*n +2232*n^2 +15276*n^3, {n,40}] (* G. C. Greubel, Sep 01 2019 *)
  • PARI
    vector(40, n, 1 +85*n +2232*n^2 +15276*n^3) \\ G. C. Greubel, Sep 01 2019
  • Sage
    [1 +85*n +2232*n^2 +15276*n^3 for n in (0..40)] # G. C. Greubel, Sep 01 2019

Formula

G.f.: x*(17594 + 60931*x + 13132*x^2 - x^3)/(1-x)^4 . - R. J. Mathar, Sep 02 2011
a(n) = +4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - R. J. Mathar, Sep 02 2011
E.g.f.: (1 + 17593*x + 48060*x^2 + 15276*x^3)*exp(x) -1. - G. C. Greubel, Apr 09 2016

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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