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 A165808 Expansion of x*(403+2967*x+1047*x^2-x^3)/(1-x)^4. 10

%I

%S 403,4579,16945,41917,83911,147343,236629,356185,510427,703771,940633,

%T 1225429,1562575,1956487,2411581,2932273,3522979,4188115,4932097,

%U 5759341,6674263,7681279,8784805,9989257,11299051,12718603,14252329,15904645

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

%C 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.

%H G. C. Greubel, <a href="/A165808/b165808.txt">Table of n, a(n) for n = 1..5000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).

%F From _R. J. Mathar_, Sep 30 2009: (Start)

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

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

%F From _G. C. Greubel_, Apr 08 2016: (Start)

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

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

%e 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.

%t LinearRecurrence[{4, -6, 4, -1}, {403, 4579, 16945, 41917}, 100](* _G. C. Greubel_, Apr 08 2016 *)

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

%Y Cf. A165806, A165809.

%K nonn,easy

%O 1,1

%A _A.K. Devaraj_, Sep 29 2009

%E More terms from _R. J. Mathar_, Sep 30 2009

%E Edited by _Jon E. Schoenfield_, Dec 12 2013

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Last modified February 22 18:24 EST 2020. Contains 332148 sequences. (Running on oeis4.)