A165808 Expansion of x*(403+2967*x+1047*x^2-x^3)/(1-x)^4.

403, 4579, 16945, 41917, 83911, 147343, 236629, 356185, 510427, 703771, 940633, 1225429, 1562575, 1956487, 2411581, 2932273, 3522979, 4188115, 4932097, 5759341, 6674263, 7681279, 8784805, 9989257, 11299051, 12718603, 14252329, 15904645

Offset

1,1

Comments

Old name was: As mentioned in short description of A165806, polynomials have the following unique property: let f(x) be a polynomial in x. Then f(x+k*f(x)) is congruent to 0 (mod(f(x)); here k belongs to N. The present case pertains to f(x) = x^3 + 2x + 11 when x is complex (2 + 3i). The quotient f(x+k*f(x))/f(x), for any given k, consists of two parts: a) a rational integer part and b) rational integer coefficient of sqrt(-1). This sequence pertains to a.

Links

G. C. Greubel, Table of n, a(n) for n = 1..5000

Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Formula

From R. J. Mathar, Sep 30 2009: (Start)

a(n) = 1-13*n-321*n^2+736*n^3.

G.f.: x*(403+2967*x+1047*x^2-x^3)/(1-x)^4. (End)

From G. C. Greubel, Apr 08 2016: (Start)

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).

E.g.f.: (1 -333*x - 318*x^2 + x^3)*exp(x). (end)

Example

f(x)= x^3 + 2x + 11. When x = 2 + 3i, we get f(x) = -31 + 15i. x + f(x) = -29 + 18i. f(-29 + 18i) = 3752 + 39618i. When this value is divided by (-31 + 15i) we get 403 - 1083i; needless to say, PARI takes care of necessary rationalization.

Mathematica

LinearRecurrence[{4, -6, 4, -1}, {403, 4579, 16945, 41917}, 100](* G. C. Greubel, Apr 08 2016 *)

Prog

(PARI) Vec((403+2967*x+1047*x^2-x^3)/(1-x)^4+O(x^99)) \\ Charles R Greathouse IV, Sep 23 2012

Crossrefs

Cf. A165806, A165809.

Sequence in context: A083815 A250893 A261857 * A283662 A097741 A117836

Adjacent sequences: A165805 A165806 A165807 * A165809 A165810 A165811

Keyword

nonn,easy

Author

A.K. Devaraj, Sep 29 2009

Extensions

More terms from R. J. Mathar, Sep 30 2009

Edited by Jon E. Schoenfield, Dec 12 2013

Status

approved