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Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.
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%I #24 Sep 16 2024 12:52:10

%S 1,5,9,17,29,49,81,133,217,353,573,929,1505,2437,3945,6385,10333,

%T 16721,27057,43781,70841,114625,185469,300097,485569,785669,1271241,

%U 2056913,3328157,5385073,8713233,14098309,22811545,36909857,59721405,96631265

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

%C The titular polynomial is defined recursively by p(n,x)=x*(n-1,x)+3n+1 for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744.

%H G. C. Greubel, <a href="/A192746/b192746.txt">Table of n, a(n) for n = 0..1000</a>

%H Feng-Zhen Zhao, <a href="https://math.colgate.edu/~integers/y82/y82.pdf">The log-behavior of some sequences related to the generalized Leonardo numbers</a>, Integers (2024) Vol. 24, Art. No. A82.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,0,-1).

%F G.f.: (1+3*x-x^2)/((1-x)*(1-x-x^2)), so the first differences are (essentially) A022087. - _R. J. Mathar_, May 04 2014

%F a(n) = 4*Fibonacci(n+2)-3. - _Gerry Martens_, Jul 04 2015

%t (* First program *)

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

%t p[0, n_]:= 1; p[n_, x_]:= x*p[n-1, x] +3n +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}] (* A192746 *)

%t u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192747 *) (* _Clark Kimberling_, Jul 09 2011 *)

%t (* Additional programs *)

%t a[0]=1;a[1]=5;a[n_]:=a[n]=a[n-1]+a[n-2]+3;Table[a[n],{n,0,36}] (* _Gerry Martens_, Jul 04 2015 *)

%t 4*Fibonacci[Range[0,40]+2]-3 (* _G. C. Greubel_, Jul 24 2019 *)

%o (PARI) vector(30, n, n--; 4*fibonacci(n+2)-3) \\ _G. C. Greubel_, Jul 24 2019

%o (Magma) [4*Fibonacci(n+2)-3: n in [0..30]]; // _G. C. Greubel_, Jul 24 2019

%o (Sage) [4*fibonacci(n+2)-3 for n in (0..30)] # _G. C. Greubel_, Jul 24 2019

%o (GAP) List([0..30], n-> 4*Fibonacci(n+2)-3); # _G. C. Greubel_, Jul 24 2019

%Y Cf. A000045, A192232, A192744.

%K nonn

%O 0,2

%A _Clark Kimberling_, Jul 09 2011