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A192923
Coefficient of x in the reduction by (x^2->x+1) of the polynomial p(n,x) defined below at Comments.
2
0, 1, 2, 4, 9, 19, 42, 91, 200, 437, 959, 2101, 4609, 10106, 22168, 48620, 106649, 233928, 513126, 1125541, 2468901, 5415578, 11879209, 26057330, 57157443, 125376341, 275016369, 603255761
OFFSET
0,3
COMMENTS
The titular polynomial is defined by p(n,x) = p(n-1,x) +(x^2)*p(n-2,x), with p(0,x)=1, p(1,x)=x. For discussions of polynomial reduction, see A192232, A192744, and A192872.
FORMULA
a(n) = 2*a(n-1) + 2*a(n-2) - 3*a(n-3) - a(n-4).
G.f.: x*(1-2*x^2) / ( 1-2*x-2*x^2+3*x^3+x^4 ). - R. J. Mathar, May 08 2014
MATHEMATICA
(See A192922.)
CoefficientList[Series[x*(1-2*x^2)/(1-2*x-2*x^2+3*x^3+x^4), {x, 0, 30}], x] (* G. C. Greubel, Jun 26 2017 *)
PROG
(PARI) x='x+O('x^30); concat([0], Vec(x*(1-2*x^2)/(1-2*x-2*x^2+3*x^3+x^4) )) \\ G. C. Greubel, Jun 26 2017
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!( x*(1-2*x^2)/(1-2*x-2*x^2+3*x^3+x^4) )); // G. C. Greubel, Feb 06 2019
(Sage) (x*(1-2*x^2)/(1-2*x-2*x^2+3*x^3+x^4)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, Feb 06 2019
(GAP) a:=[0, 1, 2, 4];; for n in [5..30] do a[n]:=2*a[n-1]+2*a[n-2]-3*a[n-3] -a[n-4]; od; a; # G. C. Greubel, Feb 06 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Clark Kimberling, Jul 12 2011
STATUS
approved