A192960 - OEIS

%I #16 Sep 08 2022 08:45:58

%S 1,3,7,15,29,53,93,159,267,443,729,1193,1945,3163,5135,8327,13493,

%T 21853,35381,57271,92691,150003,242737,392785,635569,1028403,1664023,

%U 2692479,4356557,7049093,11405709,18454863,29860635,48315563,78176265

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

%C The titular polynomials are defined recursively:  p(n,x) = x*p(n-1,x) + 2 + n^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 G. C. Greubel, <a href="/A192960/b192960.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-1,1).

%F a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).

%F From _R. J. Mathar_, May 09 2014: (Start)

%F G.f.: (1+x)*(1-x+x^2)/((1-x-x^2)*(1-x)^2).

%F a(n) - a(n-1) = A019274(n+2). (End)

%F a(n) = 2*Fibonacci(n+4) - (2*n + 5). - _G. C. Greubel_, Jul 12 2019

%t (* First program *)

%t q = x^2; s = x + 1; z = 40;

%t p[0, x]:= 1;

%t p[n_, x_]:= x*p[n - 1, x] + n^2 + 2;

%t Table[Expand[p[n, x]], {n, 0, 7}]

%t reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]

%t t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];

%t u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192960 *)

%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192961 *)

%t (* Second program *)

%t With[{F=Fibonacci}, Table[2*F[n+4]-(2*n+5), {n,0,40}]] (* _G. C. Greubel_, Jul 12 2019 *)

%o (PARI) vector(40, n, n--; f=fibonacci; 2*f(n+4)-(2*n+5)) \\ _G. C. Greubel_, Jul 12 2019

%o (Magma) F:=Fibonacci; [2*F(n+4)-(2*n+5): n in [0..40]]; // _G. C. Greubel_, Jul 12 2019

%o (Sage) f=fibonacci; [2*f(n+4)-(2*n+5) for n in (0..40)] # _G. C. Greubel_, Jul 12 2019

%o (GAP) F:=Fibonacci;; List([0..40], n-> 2*F(n+4)-(2*n+5)); # _G. C. Greubel_, Jul 12 2019

%Y Cf. A000045, A192232, A192744, A192951, A192961.

%K nonn

%O 0,2

%A _Clark Kimberling_, Jul 13 2011