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%I #52 Sep 08 2022 08:45:18
%S 1,0,0,0,1,0,1,1,2,2,4,5,8,11,17,24,36,52,77,112,165,241,354,518,760,
%T 1113,1632,2391,3505,5136,7528,11032,16169,23696,34729,50897,74594,
%U 109322,160220,234813,344136,504355,739169,1083304,1587660,2326828,3410133,4997792,7324621
%N Expansion of g.f.: (1-x^2-x^3)/( (1+x)*(1-x-x^3) ).
%C The sequence can be interpreted as the top-left entry of the n-th power of a 4 X 4 (0,1) matrix. There are 12 different choices (out of 2^16) for that (0,1) matrix. - _R. J. Mathar_, Mar 19 2014
%H Reinhard Zumkeller, <a href="/A107458/b107458.txt">Table of n, a(n) for n = 0..1000</a>
%H C. Kenneth Fan, <a href="http://dx.doi.org/10.1090/S0894-0347-97-00222-1">Structure of a Hecke algebra quotient</a>, J. Amer. Math. Soc. 10 (1997), no. 1, 139-167. [Page 156, f^2_n.]
%H Renata Passos Machado Vieira, Francisco Regis Vieira Alves, <a href="https://doi.org/10.7546/nntdm.2019.25.3.185-197">Sequences of Tridovan and their identities</a>, Notes on Number Theory and Discrete Mathematics (2019) Vol. 25, No. 3, 185-197. Sequence (T_n) is a subsequence of this sequence.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,1,1).
%F a(n) = a(n-2) + a(n-3) + a(n-4); a(0)=1, a(1)=0, a(2)=0, a(3)=0. - _Harvey P. Dale_, Jun 20 2011
%F a(n) + a(n-1) = A000930(n-4). - _R. J. Mathar_, Mar 19 2014
%p seq(coeff(series( (1-x^2-x^3)/( (1+x)*(1-x-x^3) ), x, n+1), x, n), n = 0..50); # _G. C. Greubel_, Jan 03 2020
%t CoefficientList[Series[(1-x^2-x^3)/(1-x^2-x^3-x^4),{x,0,50}],x] (* or *) LinearRecurrence[{0,1,1,1},{1,0,0,0},50] (* _Harvey P. Dale_, Jun 20 2011 *)
%o (Haskell)
%o a107458 n = a107458_list !! n
%o a107458_list = 1 : 0 : 0 : 0 : zipWith (+) a107458_list
%o (zipWith (+) (tail a107458_list) (drop 2 a107458_list))
%o -- _Reinhard Zumkeller_, Mar 23 2012
%o (PARI) my(x='x+O('x^50)); Vec((1-x^2-x^3)/((1+x)*(1-x-x^3))) \\ _G. C. Greubel_, Apr 27 2017
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 50); Coefficients(R!(1-x^2-x^3)/( (1+x)*(1-x-x^3))); // _Marius A. Burtea_, Jan 02 2020
%o (Sage)
%o def A107458_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1-x^2-x^3)/((1+x)*(1-x-x^3)) ).list()
%o A107458_list(50) # _G. C. Greubel_, Jan 03 2020
%o (GAP) a:=[1,0,0,0];; for n in [5..50] do a[n]:=a[n-2]+a[n-3]+a[n-4]; od; a; # _G. C. Greubel_, Jan 03 2020
%Y Cf. A001634, A013979, A078012, A135851.
%K nonn,easy
%O 0,9
%A _N. J. A. Sloane_, Mar 08 2008
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