cp's OEIS Frontend

A072176 Unimodal analog of Fibonacci numbers: a(n+1) = Sum_{k=0..floor(n/2)} A071922(n-k,k).

%I A072176 #17 Sep 08 2022 08:45:06
%S A072176 1,1,2,3,5,9,16,30,56,106,201,382,727,1384,2636,5021,9565,18222,34715,
%T A072176 66137,126001,240052,457338,871304,1659978,3162533,6025150,11478911,
%U A072176 21869232,41664520,79377833,151227961,288114394,548905795
%N A072176 Unimodal analog of Fibonacci numbers: a(n+1) = Sum_{k=0..floor(n/2)} A071922(n-k,k).
%C A072176 Based on the observation that F_{n+1} = Sum_{k} binomial (n-k,k). In both cases the sum is extended to 0<=2k<=n.
%H A072176 G. C. Greubel, <a href="/A072176/b072176.txt">Table of n, a(n) for n = 1..1000</a>
%H A072176 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2,-1,1).
%F A072176 G.f.: x*(1-x-x^2)/((1-x)*(1-x-2*x^2+x^4)).
%F A072176 a(1)=1, a(2)=1, a(3)=2, a(4)=3, a(5)=5, a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3) - a(n-4) + a(n-5). - _Harvey P. Dale_, Jun 23 2011
%p A072176 seq(coeff(series(x*(1-x-x^2)/((1-x)*(1-x-2*x^2+x^4)), x, n+1), x, n), n = 1..40); # _G. C. Greubel_, Aug 26 2019
%t A072176 Rest@CoefficientList[ Series[x(1-x-x^2)/((1-x)(1-x-2x^2+x^4)), {x, 0, 40}], x] (* or *) LinearRecurrence[{2,1,-2,-1,1}, {1,1,2,3,5}, 40] (* _Harvey P. Dale_, Jun 23 2011 *)
%o A072176 (PARI) my(x='x+O('x^40)); Vec(x*(1-x-x^2)/((1-x)*(1-x-2*x^2+x^4))) \\ _G. C. Greubel_, Aug 26 2019
%o A072176 (Magma) R<x>:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x*(1-x-x^2)/((1-x)*(1-x-2*x^2+x^4)) )); // _G. C. Greubel_, Aug 26 2019
%o A072176 (Sage)
%o A072176 def A072176_list(prec):
%o A072176     P.<x> = PowerSeriesRing(ZZ, prec)
%o A072176     return P( x*(1-x-x^2)/((1-x)*(1-x-2*x^2+x^4)) ).list()
%o A072176 a=A072176_list(40); a[1:] # _G. C. Greubel_, Aug 26 2019
%o A072176 (GAP) a:=[1,1,2,3,5];; for n in [6..40] do a[n]:=2*a[n-1]+a[n-2] -2*a[n-3]-a[n-4]+a[n-5]; od; a; # _G. C. Greubel_, Aug 26 2019
%Y A072176 Cf. A071922, A005578.
%K A072176 nonn,easy
%O A072176 1,3
%A A072176 Michele Dondi (bik.mido(AT)tiscalinet.it), Jun 30 2002