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

0,2

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

Index entries for linear recurrences with constant coefficients, signature (2,0,-1).

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

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 *)

Prog

(PARI) vector(30, n, 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
(Sage) [4*fibonacci(n+2)-3 for n in (0..30)] # G. C. Greubel, Jul 24 2019
(GAP) List([0..30], n-> 4*Fibonacci(n+2)-3); # G. C. Greubel, Jul 24 2019

Crossrefs

Cf. A000045, A192232, A192744.

Sequence in context: A200078 A190806 A294774 * A081295 A180565 A233187

Adjacent sequences: A192743 A192744 A192745 * A192747 A192748 A192749

Keyword

nonn

Author

Clark Kimberling, Jul 09 2011

Status

approved