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, 17679967, 19582711

Offset: 1

Author

A.K. Devaraj, Sep 29 2009

Keywords

• nonn
• easy

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.

Examples

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

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).

Crossrefs

Cf. A165806, A165809.

Programs

Mathematica

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

PARI

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

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 + 402*x + 1887*x^2 + 736*x^3)*exp(x) - 1. [corrected by Jason Yuen, Aug 14 2025] (End)

Extensions

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

Edited by Jon E. Schoenfield, Dec 12 2013