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%I #26 Sep 08 2022 08:45:48
%S 17594,131307,432796,1013717,1965726,3380479,5349632,7964841,11317762,
%T 15500051,20603364,26719357,33939686,42356007,52059976,63143249,
%U 75697482,89814331,105585452,123102501,142457134,163741007,187045776
%N a(n) = 1 + 85*n + 2232*n^2 + 15276*n^3.
%C As mentioned in A166957, polynomials in two variables, not necessarily homogeneous, also have a property similar to that in a single variable (cf. A165806, A165808 and A165809) viz f(x+k*f(x,y), y + k*f(x,y)) is congruent to 0 (mod(f(x,y)). The quotient has two parts: a rational integer and a rational integer coefficient of sqrt(-1), when x belongs to Z(x = 5) and y is complex (sqrt(-1)). The polynomial considered is identical with that in A166957 viz x^3 + 2xy + y^2. The present sequence is only that of the rational integers and seq A167191 will consist of rational integer coefficients of sqrt(-1). Note: k belongs to N.
%H Vincenzo Librandi, <a href="/A167190/b167190.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,4,-1).
%F G.f.: x*(17594 + 60931*x + 13132*x^2 - x^3)/(1-x)^4 . - _R. J. Mathar_, Sep 02 2011
%F a(n) = +4*a(n-1) -6*a(n-2) +4*a(n-3) -a(n-4). - _R. J. Mathar_, Sep 02 2011
%F E.g.f.: (1 + 17593*x + 48060*x^2 + 15276*x^3)*exp(x) -1. - _G. C. Greubel_, Apr 09 2016
%e When x = 5 and y = i, f(x,y) = x^3 + 2xy + y^2 = 124 + 10i. The quotient of f(x + f(x,y), y + f(x,y))/(124 + 10i) is 17594 + 2664i.
%p seq(1 + 85*n + 2232*n^2 + 15276*n^3, n=1..40); # _G. C. Greubel_, Sep 01 2019
%t CoefficientList[Series[(17594+60931*x+13132*x^2-x^3)/(x-1)^4,{x,0,40}],x] (* _Vincenzo Librandi_, Jul 02 2012 *)
%t Table[1 +85*n +2232*n^2 +15276*n^3, {n,40}] (* _G. C. Greubel_, Sep 01 2019 *)
%o (Magma) I:=[17594, 131307, 432796, 1013717]; [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 02 2012
%o (PARI) vector(40, n, 1 +85*n +2232*n^2 +15276*n^3) \\ _G. C. Greubel_, Sep 01 2019
%o (Sage) [1 +85*n +2232*n^2 +15276*n^3 for n in (0..40)] # _G. C. Greubel_, Sep 01 2019
%o (GAP) List([0..40], n-> 1 +85*n +2232*n^2 +15276*n^3); # _G. C. Greubel_, Sep 01 2019
%Y Cf. A165806, A165808, A165809, A166957.
%K nonn,easy
%O 1,1
%A _A.K. Devaraj_, Oct 30 2009
%E Extended beyond a(6) by _R. J. Mathar_, Nov 17 2009
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