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A192953 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

%I A192953 #19 Sep 08 2022 08:45:58
%S A192953 0,1,2,6,13,26,48,85,146,246,409,674,1104,1801,2930,4758,7717,12506,
%T A192953 20256,32797,53090,85926,139057,225026,364128,589201,953378,1542630,
%U A192953 2496061,4038746,6534864,10573669,17108594,27682326,44790985,72473378
%N A192953 Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
%C A192953 The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 2n - 1, 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 A192953 G. C. Greubel, <a href="/A192953/b192953.txt">Table of n, a(n) for n = 0..1000</a>
%H A192953 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F A192953 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
%F A192953 G.f.: x*(1 -x +2*x^2)/((1-x-x^2)*(1-x)^2). - _R. J. Mathar_, Aug 01 2011
%F A192953 a(n) = -2*n - 3 + 3*A000045(n+2). - _R. J. Mathar_, Aug 01 2011
%F A192953 a(n) = A131300(n) - 1. - _R. J. Mathar_, Mar 24 2018
%F A192953 a(n) = 3*Fibonacci(n+2) - (2*n+3). - _G. C. Greubel_, Jul 12 2019
%t A192953 (* First program *)
%t A192953 q = x^2; s = x + 1; z = 40;
%t A192953 p[0, x]:= 1;
%t A192953 p[n_, x_]:= x*p[n-1, x] + 2n - 1;
%t A192953 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192953 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192953 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192953 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A111314 *)
%t A192953 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192953 *)
%t A192953 (* Second program *)
%t A192953 With[{F=Fibonacci}, Table[3*F[n+2]-(2*n+3), {n,0,40}]] (* _G. C. Greubel_, Jul 12 2019 *)
%o A192953 (PARI) vector(40, n, n--; f=fibonacci; 3*f(n+2)-(2*n+3)) \\ _G. C. Greubel_, Jul 12 2019
%o A192953 (Magma) F:=Fibonacci; [3*F(n+2)-(2*n+3): n in [0..40]]; // _G. C. Greubel_, Jul 12 2019
%o A192953 (Sage) f=fibonacci; [3*f(n+2)-(2*n+3) for n in (0..40)] # _G. C. Greubel_, Jul 12 2019
%o A192953 (GAP) F:=Fibonacci;; List([0..40], n-> 3*F(n+2)-(2*n+3)); # _G. C. Greubel_, Jul 12 2019
%Y A192953 Cf. A000045, A192232, A192744, A192951.
%K A192953 nonn,easy
%O A192953 0,3
%A A192953 _Clark Kimberling_, Jul 13 2011