cp's OEIS Frontend

A192978 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

%I A192978 #40 Dec 24 2023 12:08:36
%S A192978 0,1,4,12,29,62,122,227,406,706,1203,2020,3356,5533,9072,14816,24129,
%T A192978 39218,63654,103215,167250,270886,438599,709992,1149144,1859737,
%U A192978 3009532,4869972,7880261,12751046,20632178,33384155,54017326,87402538
%N A192978 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
%C A192978 The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 1 + n + n^2, with p(0,x)=1.  For an introduction to reductions of polynomials by substitutions such as x^2 -> x+1, see A192232 and A192744.
%C A192978 Define a triangle by T(n,0) = n*(n+1) + 1, T(n,n)=1, and T(r,c) = T(r-1,c-1) + T(r-2,c-1). The sum of the terms in row(n) is a(n+1). - _J. M. Bergot_, Apr 14 2013
%H A192978 Vincenzo Librandi, <a href="/A192978/b192978.txt">Table of n, a(n) for n = 0..1000</a>
%H A192978 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).
%F A192978 a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
%F A192978 G.f.: x*(1+x^2)/((1-x-x^2)*(1-x)^3). - _R. J. Mathar_, May 12 2014
%F A192978 a(n) = Lucas(n+5) - n*(n+5) - 11. - _Ehren Metcalfe_, Jul 13 2019
%F A192978 From _Stefano Spezia_, Jul 13 2019: (Start)
%F A192978 a(n) = (1/2)*(-22 + (11 - 5*sqrt(5))*((1/2)*(1 - sqrt(5)))^n + 11*((1/2)* (1 + sqrt(5)))^n + 5*sqrt(5)*((1/2)*(1 + sqrt(5)))^n - 10*n - 2*n^2).
%F A192978 E.g.f.: (1/2)*(2 + sqrt(5))*((-47 + 21*sqrt(5))*exp(-(1/2)*(-1 + sqrt(5))*x) + (3 + sqrt(5))*exp((1/2)*(1 + sqrt(5))*x) - 2*(-2 + sqrt(5))*exp(x)*(11 + 6*x + x^2)).
%F A192978 (End)
%t A192978 (* First program *)
%t A192978 q = x^2; s = x + 1; z = 40;
%t A192978 p[0, x]:= 1;
%t A192978 p[n_, x_]:= x*p[n-1, x] +n^2 +n +1;
%t A192978 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192978 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192978 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192978 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A027181 *)
%t A192978 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192978 *)
%t A192978 (* Additional programs *)
%t A192978 CoefficientList[Series[x*(1+x^2)/((1-x-x^2)*(1-x)^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, May 13 2014 *)
%t A192978 Table[LucasL[n+5] -(n^2+5*n+11), {n,0,40}] (* _G. C. Greubel_, Jul 24 2019 *)
%t A192978 LinearRecurrence[{4,-5,1,2,-1},{0,1,4,12,29},40] (* _Harvey P. Dale_, Dec 24 2023 *)
%o A192978 (PARI) vector(40, n, n--; f=fibonacci; f(n+6)+f(n+4) -(n^2+5*n+11)) \\ _G. C. Greubel_, Jul 24 2019
%o A192978 (Magma) [Lucas(n+5)-(n^2+5*n+11): n in [0..40]]; // _G. C. Greubel_, Jul 24 2019
%o A192978 (Sage) [lucas_number2(n+5, 1,-1) -(n^2+5*n+11) for n in (0..40)] # _G. C. Greubel_, Jul 24 2019
%o A192978 (GAP) List([0..40], n-> Lucas(1,-1, n+5)[2] -(n^2+5*n+11)); # _G. C. Greubel_, Jul 24 2019
%Y A192978 Cf. A000032, A000045, A192232, A192744, A192951.
%K A192978 nonn
%O A192978 0,3
%A A192978 _Clark Kimberling_, Jul 13 2011