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A192967 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

%I A192967 #30 Sep 08 2022 08:45:58
%S A192967 1,0,2,4,9,17,31,54,92,154,255,419,685,1116,1814,2944,4773,7733,12523,
%T A192967 20274,32816,53110,85947,139079,225049,364152,589226,953404,1542657,
%U A192967 2496089,4038775,6534894,10573700,17108626,27682359,44791019,72473413,117264468
%N A192967 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
%C A192967 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 A192967 Vincenzo Librandi, <a href="/A192967/b192967.txt">Table of n, a(n) for n = 0..1000</a>
%H A192967 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F A192967 a(0)=1, a(1)=0, for n > 1, a(n) = a(n-1) + a(n-2) + n - 1. - _Alex Ratushnyak_, May 10 2012
%F A192967 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
%F A192967 G.f.: (1 -3*x +4*x^2 -x^3)/((1-x-x^2)*(1-x)^2). - _R. J. Mathar_, May 11 2014
%F A192967 a(n) = 3*Fibonacci(n+1) - n - 2. - _G. C. Greubel_, Jul 11 2019
%t A192967 q = x^2; s = x + 1; z = 40;
%t A192967 p[0, x]:= 1;
%t A192967 p[n_, x_]:= x*p[n-1, x] + n*(n-1)/2;
%t A192967 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192967 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192967 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192967 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192967 *)
%t A192967 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192968 *)
%t A192967 LinearRecurrence[{3,-2,-1,1}, {1,0,2,4}, 41] (* _Vladimir Joseph Stephan Orlovsky_, Feb 15 2012 *)
%t A192967 Table[3*Fibonacci[n+1] -n-2, {n,0,40}] (* _G. C. Greubel_, Jul 11 2019 *)
%o A192967 (Magma) I:=[1, 0, 2, 4]; [n le 4 select I[n] else 3*Self(n-1)-2*Self(n-2)-Self(n-3)+Self(n-4): n in [1..40]]; // _Vincenzo Librandi_, Feb 20 2012
%o A192967 (Magma) [3*Fibonacci(n+1) -n-2: n in [0..40]]; // _G. C. Greubel_, Jul 11 2019
%o A192967 (PARI) vector(40, n, n--; f=fibonacci; 3*f(n+1)-n-2) \\ _G. C. Greubel_, Jul 11 2019
%o A192967 (Sage) [3*fibonacci(n+1) -n-2 for n in (0..40)] # _G. C. Greubel_, Jul 11 2019
%o A192967 (GAP) List([0..40], n-> 3*Fibonacci(n+1) -n-2); # _G. C. Greubel_, Jul 11 2019
%Y A192967 Cf. A000045, A192232, A192744, A192951.
%K A192967 nonn,easy
%O A192967 0,3
%A A192967 _Clark Kimberling_, Jul 13 2011