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A192968 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

%I A192968 #18 Sep 08 2022 08:45:58
%S A192968 0,1,1,3,7,16,33,64,118,210,364,619,1038,1723,2839,4653,7597,12370,
%T A192968 20103,32626,52900,85716,138826,224773,363852,588901,953053,1542279,
%U A192968 2495683,4038340,6534429,10573204,17108098,27681798,44790424,72472783
%N A192968 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
%C A192968 The titular polynomials are defined recursively:  p(n,x) = x*p(n-1,x) + n(n-1)/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.
%H A192968 G. C. Greubel, <a href="/A192968/b192968.txt">Table of n, a(n) for n = 0..1000</a>
%H A192968 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (4,-5,1,2,-1).
%F A192968 a(n) = 4*a(n-1) - 5*a(n-2) + a(n-3) + 2*a(n-4) - a(n-5).
%F A192968 G.f.: x*(1 -3*x +4*x^2 -x^3)/((1-x-x^2)*(1-x)^3). - _R. J. Mathar_, May 11 2014
%F A192968 a(n) = 3*Fibonacci(n+2) -(n^2+3*n+6)/2. - _G. C. Greubel_, Jul 11 2019
%t A192968 Table[3*Fibonacci[n+2] -(n^2+3*n+6)/2, {n, 0, 40}] (* _G. C. Greubel_, Jul 11 2019 *)
%o A192968 (PARI) vector(40, n, n--; 3*fibonacci(n+2) -(n^2+3*n+6)/2) \\ _G. C. Greubel_, Jul 11 2019
%o A192968 (Magma) [3*Fibonacci(n+2) -(n^2+3*n+6)/2: n in [0..40]]; // _G. C. Greubel_, Jul 11 2019
%o A192968 (Sage) [3*fibonacci(n+2) -(n^2+3*n+6)/2 for n in (0..40)] # _G. C. Greubel_, Jul 11 2019
%o A192968 (GAP) List([0..40], n-> 3*Fibonacci(n+2) -(n^2+3*n+6)/2); # _G. C. Greubel_, Jul 11 2019
%Y A192968 Cf. A000045, A192232, A192744, A192951, A192967.
%K A192968 nonn
%O A192968 0,4
%A A192968 _Clark Kimberling_, Jul 13 2011