A192880 Constant term in the reduction by (x^2 -> x + 1) of the polynomial p(n,x) given in Comments.
1, 0, 3, 7, 34, 123, 495, 1912, 7501, 29253, 114342, 446545, 1744489, 6814224, 26618619, 103979239, 406172770, 1586623227, 6197795703, 24210320296, 94572284197, 369425778645, 1443080391558, 5637075481729, 22019992977457
Offset: 0
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (2,7,2,-1).
Programs
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GAP
a:=[1,0,3,7];; for n in [5..30] do a[n]:=2*a[n-1]+7*a[n-2] +2*a[n-3] -a[n-4]; od; a; # G. C. Greubel, Jan 08 2019
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Magma
m:=30; R
:=PowerSeriesRing(Integers(), m); Coefficients(R!( (1+x)*(1-3*x-x^2)/(1-2*x-7*x^2-2*x^3+x^4) )); // G. C. Greubel, Jan 08 2019 -
Mathematica
(* First program *) q = x^2; s = x + 1; z = 25; p[0, x_]:= 1; p[1, x_]:= x; p[n_, x_]:= 2 p[n-1, x]*x + p[n-2, x]*x^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}] (* A192880 *) u2 = Table[Coefficient[Part[t, n], x, 1], {n, 1, z}] (* A192882 *) FindLinearRecurrence[u1] FindLinearRecurrence[u2] (* Additional programs *) LinearRecurrence[{2,7,2,-1}, {1,0,3,7}, 30] (* G. C. Greubel, Jan 08 2019 *) Table[Fibonacci[n-1]*LucasL[n, 2]/2, {n,0,30}] (* G. C. Greubel, Jul 29 2019 *) -
PARI
my(x='x+O('x^30)); Vec((1+x)*(1-3*x-x^2)/(1-2*x-7*x^2-2*x^3+x^4)) \\ G. C. Greubel, Jan 08 2019 -
SageMath
((1+x)*(1-3*x-x^2)/(1-2*x-7*x^2-2*x^3+x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 08 2019
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SageMath
[fibonacci(n-1)*lucas_number2(n, 2, -1)/2 for n in (0..30)] # G. C. Greubel, Jul 29 2019
Formula
a(n) = 2*a(n-1) + 7*a(n-2) + 2*a(n-3) - a(n-4).
G.f.: (1+x)*(1-3*x-x^2) / ( 1-2*x-7*x^2-2*x^3+x^4 ). - R. J. Mathar, May 07 2014
a(n) = Fibonacci(n-1)*Pell-Lucas(n)/2, where Pell-Lucas(n) = A002203(n). - G. C. Greubel, Jul 29 2019
Comments