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A192952
Coefficient of x in the reduction by x^2 -> x+1 of the polynomial p(n,x) defined at Comments.
2
0, 1, 2, 7, 16, 33, 62, 111, 192, 325, 542, 895, 1468, 2397, 3902, 6339, 10284, 16669, 27002, 43723, 70780, 114561, 185402, 300027, 485496, 785593, 1271162, 2056831, 3328072, 5384985, 8713142, 14098215, 22811448, 36909757, 59721302, 96631159
OFFSET
0,3
COMMENTS
The titular polynomials are defined recursively: p(n,x) = x*p(n-1,x) + 3n - 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.
FORMULA
a(n) = 3*a(n-1) - 2*a(n-2) - a(n-3) + a(n-4).
From R. J. Mathar, May 08 2014: (Start)
G.f.: x*(1 -x +3*x^2)/((1-x-x^2)*(1-x)^2).
a(n) - a(n-1) = A192746(n-2). (End)
a(n) = 4*Fibonacci(n+2) - (3*n+4). - G. C. Greubel, Jul 12 2019
MATHEMATICA
(* First program *)
q = x^2; s = x + 1; z = 40;
p[0, x]:= 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}] (* A192952 *)
(* Second program *)
With[{F=Fibonacci}, Table[4*F[n+2]-(3*n+4), {n, 0, 40}]] (* G. C. Greubel, Jul 12 2019 *)
PROG
(PARI) vector(40, n, n--; f=fibonacci; 4*f(n+2)-(3*n+4)) \\ G. C. Greubel, Jul 12 2019
(Magma) F:=Fibonacci; [4*F(n+2)-(3*n+4): n in [0..40]]; // G. C. Greubel, Jul 12 2019
(Sage) f=fibonacci; [4*f(n+2)-(3*n+4) for n in (0..40)] # G. C. Greubel, Jul 12 2019
(GAP) F:=Fibonacci;; List([0..40], n-> 4*F(n+2)-(3*n+4)); # G. C. Greubel, Jul 12 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 13 2011
STATUS
approved