A192913 Coefficient of x^2 in the reduction by (x^3 -> x + 1) of the polynomial F(n+1)*x^n, where F(n)=A000045 (Fibonacci sequence).
0, 0, 2, 3, 10, 32, 91, 273, 816, 2420, 7209, 21456, 63842, 190008, 565470, 1682835, 5008192, 14904512, 44356229, 132005445, 392851940, 1169138532, 3479389655, 10354762656, 30816068600, 91709498068, 272930078466, 812247687927
Offset: 0
Keywords
Examples
(See A192911.)
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (1,4,5,2,-1,1).
Programs
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GAP
a:=[0,0,2,3,10,32];; for n in [7..30] do a[n]:=a[n-1]+4*a[n-2] +5*a[n-3]+2*a[n-4]-a[n-5]+a[n-6]; od; a; # G. C. Greubel, Jan 12 2019
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Magma
m:=30; R
:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!( x^2*(1+x)*(2-x)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6) )); // G. C. Greubel, Jan 12 2019 -
Mathematica
(See A192911.) LinearRecurrence[{1,4,5,2,-1,1},{0,0,2,3,10,32},28] (* Ray Chandler, Aug 02 2015 *)
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PARI
my(x='x+O('x^30)); concat([0,0], Vec(x^2*(1+x)*(2-x)/(1-x-4*x^2 -5*x^3-2*x^4+x^5-x^6))) \\ G. C. Greubel, Jan 12 2019 -
Sage
(x^2*(1+x)*(2-x)/(1-x-4*x^2-5*x^3-2*x^4+x^5-x^6)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Jan 12 2019
Formula
(See A192911.)
G.f.: x^2*(1+x)*(2-x) / (1 - x - 4*x^2 - 5*x^3 - 2*x^4 + x^5 - x^6). - R. J. Mathar, May 08 2014
a(n) = a(n-1) + 4*a(n-2) + 5*a(n-3) + 2*a(n-4) - a(n-5) + a(n-6). - Wesley Ivan Hurt, Aug 04 2025
Comments