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A192904
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Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) defined below at Comments.
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7
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1, 0, 1, 5, 16, 49, 153, 480, 1505, 4717, 14784, 46337, 145233, 455200, 1426721, 4471733, 14015632, 43928817, 137684905, 431542080, 1352570689, 4239325789, 13287204352, 41645725825, 130529073953, 409113752000, 1282274186177
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OFFSET
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0,4
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COMMENTS
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The titular polynomial is defined by p(n,x) = (x^2)*p(n-1,x) + x*p(n-2,x), with p(0,x) = 1, p(1,x) = x. The resulting sequence typifies a general class which we shall describe here. Suppose that u,v,a,b,c,d,e,f are numbers used to define these polynomials:
...
q(x) = x^2
s(x) = u*x + v
p(0,x) = a, p(1,x) = b*x + c
p(n,x) = d*(x^2)*p(n-1,x) + e*x*p(n-2,x) + f.
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We shall assume that u is not 0 and that {d,e} is not {0}. The reduction of p(n,x) by the repeated substitution q(x) -> s(x), as defined and described at A192232 and A192744, has the form h(n) + k(n)*x. The numerical sequences h and k are linear recurrence sequences, formally of order 5. The Mathematica program below, with first line deleted, shows initial terms and recurrence coefficients, which imply these properties:
(1) the recurrence coefficients depend only on u,v,d,e; the parameters a,b,c,f affect only the initial terms.
(2) if e=0 or v=0, the order of recurrence is <= 3;
(3) if e=0 and v=0, the recurrence coefficients are 1+d*u^2 and -d*u^2 (cf. similar results at A192872).
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Examples:
u v a b c d e f... seq h.....seq k
The terms of these sequences involve Fibonacci numbers, F(n)=A000045(n); e.g.,
A103609: (1,1,1,1,2,2,3,3,5,5,8,8,...)
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LINKS
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FORMULA
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a(n) = 3*a(n-1) + a(n-3) + a(n-4).
G.f.: (1-x)*(1-2*x-x^2)/(1-3*x-x^3-x^4). - Colin Barker, Aug 31 2012
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EXAMPLE
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The first six polynomials and reductions:
1 -> 1
x -> x
x + x^3 -> 1 + 3*x
x^2 + x^3 + x^5 -> 5 + 8*x
x^2 + 2*x^4 + x^5 + x^7 -> 16 + 25*x
x^3 + 2*x^4 + 3*x^6 + x^7 + x^9 -> 49 + 79*x, so that
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MATHEMATICA
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(* To obtain general results, delete the next line. *)
u = 1; v = 1; a = 1; b = 1; c = 0; d = 1; e = 1; f = 0;
q = x^2; s = u*x + v; z = 24;
p[0, x_] := a; p[1, x_] := b*x + c;
p[n_, x_] := d*(x^2)*p[n - 1, x] + e*x*p[n - 2, x] + f;
Table[Expand[p[n, x]], {n, 0, 8}]
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}];
u0 = Table[Coefficient[Part[t, n], x, 0], {n, 1, z}] (* A192904 *)
u1 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192905 *)
Simplify[FindLinearRecurrence[u0]] (* recurrence for 0-sequence *)
Simplify[FindLinearRecurrence[u1]] (* recurrence for 1-sequence *)
LinearRecurrence[{3, 0, 1, 1}, {1, 0, 1, 5}, 40] (* G. C. Greubel, Jan 10 2019 *)
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PROG
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(PARI) my(x='x+O('x^40)); Vec((1-x)*(1-2*x-x^2)/(1-3*x-x^3-x^4)) \\ G. C. Greubel, Jan 10 2019
(Magma) m:=40; R<x>:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1-x)*(1-2*x-x^2)/(1-3*x-x^3-x^4) )); // G. C. Greubel, Jan 10 2019
(Sage) ((1-x)*(1-2*x-x^2)/(1-3*x-x^3-x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jan 10 2019
(GAP) a:=[1, 0, 1, 5];; for n in [5..40] do a[n]:=3*a[n-1]+a[n-3]+a[n-4]; od; a; # G. C. Greubel, Jan 10 2019
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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