|
%I #19 Feb 13 2022 23:35:36
%S 1,1,4,7,18,38,88,195,441,988,2223,4992,11220,25208,56645,127277,
%T 285992,642615,1443946,3244514,7290360,16381287,36808421,82707768,
%U 185842671,417584688,938304280,2108350576,4737420745,10644887785,23918845740
%N a(n) = a(n-1) + 3*a(n-2) - a(n-4), with a(0)=1, a(1)=1, a(2)=4, a(3)=7.
%C Unsigned version of A077920.
%C The sequence is the INVERT transform of the aerated even-indexed Fibonacci numbers (i.e., of (1, 0, 3, 0, 8, 0, ...)). Sequence A131322 is the INVERT transform of the aerated odd-indexed Fibonacci numbers. - _Gary W. Adamson_, Feb 07 2014
%H G. C. Greubel, <a href="/A124400/b124400.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,0,-1).
%F G.f.: 1/(1-x-3*x^2+x^4).
%p seq(coeff(series(1/(1-x-3*x^2+x^4), x, n+1), x, n), n = 0..35); # _G. C. Greubel_, Dec 25 2019
%t LinearRecurrence[{1,3,0,-1}, {1,1,4,7}, 35] (* _G. C. Greubel_, Dec 25 2019 *)
%t CoefficientList[Series[1/(1-x-3x^2+x^4),{x,0,30}],x] (* _Harvey P. Dale_, Feb 01 2022 *)
%o (PARI) my(x='x+O('x^35)); Vec(1/(1-x-3*x^2+x^4)) \\ _G. C. Greubel_, Dec 25 2019
%o (Magma) I:=[1,1,4,7]; [n le 2 select I[n] else Self(n-1) +3*Self(n-2) -Self(n-4): n in [1..35]]; // _G. C. Greubel_, Dec 25 2019
%o (Sage)
%o def A124400_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( 1/(1-x-3*x^2+x^4) ).list()
%o A124400_list(35) # _G. C. Greubel_, Dec 25 2019
%o (GAP) a:=[1,1,4,7];; for n in [5..35] do a[n]:=a[n-1]+3*a[n-2]-a[n-4]; od; a; # _G. C. Greubel_, Dec 25 2019
%Y Cf. A131322.
%K easy,nonn
%O 0,3
%A _Philippe Deléham_, Dec 14 2006
|
