A166957 a(n) = 841*n^3 + 261*n^2 + 28*n + 1.

1, 1131, 7829, 25141, 58113, 111791, 191221, 301449, 447521, 634483, 867381, 1151261, 1491169, 1892151, 2359253, 2897521, 3512001, 4207739, 4989781, 5863173, 6832961, 7904191, 9081909, 10371161, 11776993, 13304451

Offset: 0

Author

A.K. Devaraj, Oct 25 2009

Keywords

• nonn
• easy

Comments

Polynomials in one variable have a certain property viz f(x+f(x)) == 0 (mod f(x)). This is true even when the polynomial is in two variables (not necessarily homogeneous). This sequence is a demonstration when the polynomial is x^3 + 2*x*y + y^2 (x = 2, y=3).

When x = 2 and y=3, f(x,y) = 29. Hence f((2 + 29), (3 + 29))/29 = 1131.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A165806, A165808, A165809.

Programs

GAP

List([0..40], n-> 1 +28*n +261*n^2 +841*n^3); # G. C. Greubel, Sep 02 2019

Magma

I:=[1131, 7829, 25141, 58113]; [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 01 2012

Magma

[1 +28*n +261*n^2 +841*n^3: n in [0..40]]; // G. C. Greubel, Sep 02 2019

Maple

seq(1 +28*n +261*n^2 +841*n^3, n=0..40); # G. C. Greubel, Sep 02 2019

Mathematica

CoefficientList[Series[(1131+3305*x+611*x^2-x^3)/(x-1)^4,{x,0,40}],x] (* Vincenzo Librandi, Jul 01 2012 *)
Table[841n^3+261n^2+28n+1,{n,30}] (* or *) LinearRecurrence[{4,-6,4,-1},{1131,7829,25141,58113},30] (* Harvey P. Dale, Apr 11 2013 *)

PARI

a(n) = ((2+n*29)^3 + 2*(2+n*29)*(3+n*29) + (3+n*29)^2)/29

Sage

[1 +28*n +261*n^2 +841*n^3 for n in (0..40)] # G. C. Greubel, Sep 02 2019

Formula

a(n) = 841*n^3 + 261*n^2 + 28*n + 1.

G.f.: x*(1131 + 3305*x + 611*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). - Vincenzo Librandi, Jul 01 2012

E.g.f.: (1 + 1130*x + 2784*x^2 + 841*x^3)*exp(x). - G. C. Greubel, Apr 09 2016

Extensions

Formula, description, editing, and program correction by Charles R Greathouse IV, Nov 04 2009

a(0)=1 added by N. J. A. Sloane, Apr 09 2016