This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A165808 #25 Aug 15 2025 00:58:21
%S A165808 403,4579,16945,41917,83911,147343,236629,356185,510427,703771,940633,
%T A165808 1225429,1562575,1956487,2411581,2932273,3522979,4188115,4932097,
%U A165808 5759341,6674263,7681279,8784805,9989257,11299051,12718603,14252329,15904645,17679967,19582711
%N A165808 Expansion of x*(403+2967*x+1047*x^2-x^3)/(1-x)^4.
%C A165808 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 A165808 G. C. Greubel, <a href="/A165808/b165808.txt">Table of n, a(n) for n = 1..5000</a>
%H A165808 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F A165808 From _R. J. Mathar_, Sep 30 2009: (Start)
%F A165808 a(n) = 1-13*n-321*n^2+736*n^3.
%F A165808 G.f.: x*(403+2967*x+1047*x^2-x^3)/(1-x)^4. (End)
%F A165808 From _G. C. Greubel_, Apr 08 2016: (Start)
%F A165808 a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4).
%F A165808 E.g.f.: (1 + 402*x + 1887*x^2 + 736*x^3)*exp(x) - 1. [corrected by _Jason Yuen_, Aug 14 2025] (End)
%e A165808 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.
%t A165808 LinearRecurrence[{4, -6, 4, -1}, {403, 4579, 16945, 41917}, 100](* _G. C. Greubel_, Apr 08 2016 *)
%o A165808 (PARI) Vec((403+2967*x+1047*x^2-x^3)/(1-x)^4+O(x^99)) \\ _Charles R Greathouse IV_, Sep 23 2012
%Y A165808 Cf. A165806, A165809.
%K A165808 nonn,easy
%O A165808 1,1
%A A165808 _A.K. Devaraj_, Sep 29 2009
%E A165808 More terms from _R. J. Mathar_, Sep 30 2009
%E A165808 Edited by _Jon E. Schoenfield_, Dec 12 2013