

A192746


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


5



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

0,2


COMMENTS

The titular polynomial is defined recursively by p(n,x)=x*(n1,x)+3n+1 for n>0, where p(0,x)=1. For discussions of polynomial reduction, see A192232 and A192744.


LINKS



FORMULA

G.f.: (1+3*xx^2)/((1x)*(1xx^2)), so the first differences are (essentially) A022087.  R. J. Mathar, May 04 2014


MATHEMATICA

(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, n_]:= 1; p[n_, x_]:= x*p[n1, 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 *)
(* Additional programs *)
a[0]=1; a[1]=5; a[n_]:=a[n]=a[n1]+a[n2]+3; Table[a[n], {n, 0, 36}] (* Gerry Martens, Jul 04 2015 *)


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



KEYWORD

nonn


AUTHOR



STATUS

approved



