OFFSET
0,2
COMMENTS
Polynomials in one variable have a certain property viz f(x+f(x)) is congruent to 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).
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
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 *)
PROG
(PARI) a(n) = ((2+n*29)^3 + 2*(2+n*29)*(3+n*29) + (3+n*29)^2)/29
(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
(Sage) [1 +28*n +261*n^2 +841*n^3 for n in (0..40)] # G. C. Greubel, Sep 02 2019
(GAP) List([0..40], n-> 1 +28*n +261*n^2 +841*n^3); # G. C. Greubel, Sep 02 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
A.K. Devaraj, Oct 25 2009
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
STATUS
approved