A192746 Constant term of the reduction by x^2 -> x+1 of the polynomial p(n,x) defined below in Comments.

1, 5, 9, 17, 29, 49, 81, 133, 217, 353, 573, 929, 1505, 2437, 3945, 6385, 10333, 16721, 27057, 43781, 70841, 114625, 185469, 300097, 485569, 785669, 1271241, 2056913, 3328157, 5385073, 8713233, 14098309, 22811545, 36909857, 59721405, 96631265

Offset: 0

Author

Clark Kimberling, Jul 09 2011

Keywords

• nonn

Comments

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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Feng-Zhen Zhao, The log-behavior of some sequences related to the generalized Leonardo numbers, Integers (2024) Vol. 24, Art. No. A82.
• Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

Crossrefs

Cf. A000045, A192232, A192744.

Programs

GAP

List([0..30], n-> 4*Fibonacci(n+2)-3); # G. C. Greubel, Jul 24 2019

Magma

[4*Fibonacci(n+2)-3: n in [0..30]]; // G. C. Greubel, Jul 24 2019

Mathematica

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, n_]:= 1; p[n_, x_]:= x*p[n-1, x] +3n +2;
Table[Expand[p[n, x]], {n, 0, 7}]
reduce[{p1_, q_, s_, x_}]:= FixedPoint[(s PolynomialQuotient @@ #1 + PolynomialRemainder @@ #1 &)[{#1, q, x}] &, p1]
t = Table[reduce[{p[n, x], q, s, x}], {n, 0, z}];
u1 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192746 *)
u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192747 *) (* Clark Kimberling, Jul 09 2011 *)
(* Additional programs *)
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 *)
4*Fibonacci[Range[0,40]+2]-3 (* G. C. Greubel, Jul 24 2019 *)

PARI

vector(30, n, n--; 4*fibonacci(n+2)-3) \\ G. C. Greubel, Jul 24 2019

Sage

[4*fibonacci(n+2)-3 for n in (0..30)] # G. C. Greubel, Jul 24 2019

Formula

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

a(n) = 4*Fibonacci(n+2)-3. - Gerry Martens, Jul 04 2015