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

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

Author

A.K. Devaraj, Oct 30 2009

Keywords

• nonn
• easy

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

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

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