Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%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