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A192969 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.

%I A192969 #32 Sep 08 2022 08:45:58
%S A192969 1,2,6,12,23,41,71,120,200,330,541,883,1437,2334,3786,6136,9939,16093,
%T A192969 26051,42164,68236,110422,178681,289127,467833,756986,1224846,1981860,
%U A192969 3206735,5188625,8395391,13584048,21979472,35563554,57543061,93106651
%N A192969 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
%C A192969 The titular polynomials are defined recursively:  p(n,x) = x*p(n-1,x) + n(n+3)/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 A192969 Daniel Suteu, <a href="/A192969/b192969.txt">Table of n, a(n) for n = 0..999</a>
%H A192969 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).
%F A192969 a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
%F A192969 G.f.: (1 - x + 2*x^2 - x^3)/((1-x-x^2)*(1-x)^2). - _R. J. Mathar_, May 11 2014
%F A192969 a(0) = 1; a(1) = 2; a(n) = 1 + n + a(n-1) + a(n-2). - _Daniel Suteu_, Jan 12 2016
%F A192969 a(n) = 2*Fibonacci(n+2) + 3*Fibonacci(n+1) - n - 4. - _G. C. Greubel_, Jul 11 2019
%p A192969 F:= gfun:-rectoproc({a(0) = 1, a(1) = 2, a(n) = 1 + n + a(n-1) + a(n-2)},a(n),remember):
%p A192969 map(F, [$0..100]); # _Robert Israel_, Jan 18 2016
%t A192969 (* First progream *)
%t A192969 q = x^2; s = x + 1; z = 40;
%t A192969 p[0, x] := 1;
%t A192969 p[n_, x_] := x*p[n - 1, x] + n (n + 3)/2;
%t A192969 Table[Expand[p[n, x]], {n, 0, 7}]
%t A192969 reduce[{p1_, q_, s_, x_}] :=
%t A192969 FixedPoint[(s PolynomialQuotient @@ #1 +PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
%t A192969 t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
%t A192969 u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192969 *)
%t A192969 u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192970 *)
%t A192969 (* Second program *)
%t A192969 Table[2*Fibonacci[n+2]+3*Fibonacci[n+1]-n-4, {n,0,40}] (* _G. C. Greubel_, Jul 11 2019 *)
%o A192969 (Sidef)
%o A192969 func a((0)) { 1 }
%o A192969 func a((1)) { 2 }
%o A192969 func a(n) is cached { 1 + n + a(n-1) + a(n-2) }
%o A192969 100.times { |i| say a(i-1) }
%o A192969 # _Daniel Suteu_, Jan 12 2016
%o A192969 (PARI) vector(40, n, n--; f=fibonacci; 2*f(n+2)+3*f(n+1)-n-4) \\ _G. C. Greubel_, Jul 11 2019
%o A192969 (Magma) F:=Fibonacci; [2*F(n+2)+3*F(n+1)-n-4: n in [0..40]]; // _G. C. Greubel_, Jul 11 2019
%o A192969 (Sage) f=fibonacci; [2*f(n+2)+3*f(n+1)-n-4 for n in (0..40)] # _G. C. Greubel_, Jul 11 2019
%o A192969 (GAP) F:=Fibonacci;; List([0..40], n-> 2*F(n+2)+3*F(n+1)-n-4); # _G. C. Greubel_, Jul 11 2019
%Y A192969 Cf. A000045, A192232, A192744, A192951, A192970.
%K A192969 nonn
%O A192969 0,2
%A A192969 _Clark Kimberling_, Jul 13 2011