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A192908 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.

%I A192908 #23 Sep 08 2022 08:45:58
%S A192908 1,1,3,7,17,43,111,289,755,1975,5169,13531,35423,92737,242787,635623,
%T A192908 1664081,4356619,11405775,29860705,78176339,204668311,535828593,
%U A192908 1402817467,3672623807,9615053953,25172538051,65902560199
%N A192908 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.
%C A192908 The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x), with p(0,x) = 1, p(1,x) = x + 1.
%H A192908 G. C. Greubel, <a href="/A192908/b192908.txt">Table of n, a(n) for n = 0..1000</a>
%H A192908 <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (4,-4,1).
%F A192908 a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3) for n>3.
%F A192908 G.f.: 1 + x*(1 - x - x^2)/((1 - x)*(1 - 3*x + x^2)). - _R. J. Mathar_, Jul 13 2011
%F A192908 a(n) = 2*Fibonacci(2*n-2) + 1 for n>0, a(0)=1. - _Bruno Berselli_, Dec 27 2016
%F A192908 a(n) = -1 + 3*a(n-1) - a(n-2) with a(1) = 1 and a(2) = 3. Cf. A055588 and A097136. - _Peter Bala_, Nov 12 2017
%t A192908 u = 1; v = 1; a = 1; b = 1; c = 1; d = 1; e = 0; f = 1;
%t A192908 q = x^2; s = u*x + v; z = 26;
%t A192908 p[0, x_] := a;  p[1, x_] := b*x + c
%t A192908 p[n_, x_] := d*(x^2)*p[n - 1, x] + e*x*p[n - 2, x] + f;
%t A192908 Table[Expand[p[n, x]], {n, 0, 8}]
%t A192908 reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192908 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192908 u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}]    (* A192908 *)
%t A192908 u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}]    (* A069403 *)
%t A192908 Simplify[FindLinearRecurrence[u0]] (* recurrence for 0-sequence *)
%t A192908 Simplify[FindLinearRecurrence[u1]] (* recurrence for 1-sequence *)
%t A192908 LinearRecurrence[{4,-4,1}, {1,1,3,7}, 30] (* _G. C. Greubel_, Jan 11 2019 *)
%o A192908 (PARI) vector(30, n, n--; if(n==0,1,1+2*fibonacci(2*n-2))) \\ _G. C. Greubel_, Jan 11 2019
%o A192908 (Magma) [1] cat [1+2*Fibonacci(2*(n-1)): n in [1..30]]; // _G. C. Greubel_, Jan 11 2019
%o A192908 (Sage) [1]+[1+2*fibonacci(2*(n-1)) for n in (1..30)] # _G. C. Greubel_, Jan 11 2019
%o A192908 (GAP) Concatenation([1], List([1..30], n -> 1+2*Fibonacci(2*(n-1)))); # _G. C. Greubel_, Jan 11 2019
%Y A192908 Cf. A192232, A192744, A192872, A192904.
%Y A192908 Cf. A000045; A052995: 2*Fibonacci(2*n-1) for n>0.
%Y A192908 CF. A055588, A097136.
%K A192908 nonn,easy
%O A192908 0,3
%A A192908 _Clark Kimberling_, Jul 12 2011