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%I #40 Feb 28 2023 12:07:15
%S 1,1,1,4,4,7,16,19,37,67,94,178,295,460,829,1345,2209,3832,6244,10459,
%T 17740,29191,49117,82411,136690,229762,383923,639832,1073209,1791601,
%U 2992705,5011228,8367508,13989343,23401192,39091867,65369221,109295443
%N Expansion of (1+x)/(1-x^2-3*x^3).
%H Vincenzo Librandi, <a href="/A238389/b238389.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,3).
%F a(0)=1, a(1)=1, a(2)=1; for n>2, a(n) = a(n-2) + 3*a(n-3).
%F a(2n) = Sum_{j=0}^{n/3} binomial(n-j,2j)*3^(2j) + Sum_{j=0}^{(n-2)/3} binomial(n-1-j,2j+1)*3^(2j+1).
%F a(2n+1) = Sum_{j=0}^{n/3} binomial(n-j,2j)*3^(2j) + Sum_{j=0}^{(n-1)/3} binomial(n-j,2j+1)*3^(2j+1).
%F a(n) = |A106855(n)| + |A106855(n-1)| . - _R. J. Mathar_, Mar 13 2014
%e a(3) = 3*a(0)+a(1) = 4; a(4) = 3*a(1)+a(2) = 4; a(5) = 3*a(2)+a(3) = 7.
%p a:= n-> (<<0|1|0>, <0|0|1>, <3|1|0>>^n.<<(1$3)>>)[(1$2)]:
%p seq(a(n), n=0..44); # _Alois P. Heinz_, May 09 2021
%t (* First program *)
%t For[j=0, j<3, j++, a[j] = 1]
%t For[j=3, j<51, j++, a[j] = 3a[j-3] + a[j-2]]
%t Table[a[j], {j, 0, 50}]
%t (* Second program *)
%t CoefficientList[Series[(1+x)/(1-x^2-3x^3), {x, 0, 40}], x] (* _Vincenzo Librandi_, Mar 16 2014 *)
%t LinearRecurrence[{0,1,3},{1,1,1},40] (* _Harvey P. Dale_, Feb 28 2023 *)
%o (PARI) Vec((1+x)/(1-x^2-3*x^3)+O(x^99)) \\ _Charles R Greathouse IV_, Mar 06 2014
%o (Magma) [n le 3 select 1 else Self(n-2) +3*Self(n-3): n in [1..41]]; // _G. C. Greubel_, May 09 2021
%o (Sage)
%o def A238389_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1+x)/(1-x^2-3*x^3) ).list()
%o A238389_list(40) # _G. C. Greubel_, May 09 2021
%Y Cf. A006190, A134816, A159284, A213713.
%K nonn,easy
%O 0,4
%A _Sergio Falcon_, Feb 26 2014
%E Terms corrected by _Charles R Greathouse IV_, Mar 06 2014
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