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A017817 a(n) = a(n-3) + a(n-4), with a(0)=1, a(1)=a(2)=0, a(3)=1.

%I A017817 #60 Sep 11 2024 00:48:27
%S A017817 1,0,0,1,1,0,1,2,1,1,3,3,2,4,6,5,6,10,11,11,16,21,22,27,37,43,49,64,
%T A017817 80,92,113,144,172,205,257,316,377,462,573,693,839,1035,1266,1532,
%U A017817 1874,2301,2798,3406,4175,5099,6204,7581,9274,11303,13785,16855,20577,25088
%N A017817 a(n) = a(n-3) + a(n-4), with a(0)=1, a(1)=a(2)=0, a(3)=1.
%C A017817 Number of permutations satisfying -k<=p(i)-i<=r and p(i)-i not in I, i=1..n, with k=1, r=3, I={0,1}. - _Vladimir Baltic_, Mar 07 2012
%C A017817 Number of compositions (ordered partitions) of n into parts 3 and 4.
%C A017817 For n>=2, a(n-2) is the number of ways to tile the 1xn board with dominoes and squares (ie. monominoes) such that there are either one or two squares between dominoes, no squares at either end of the board, and there is at least one domino. - _Enrique Navarrete_, Sep 01 2024
%C A017817 For n>=3, a(n-3) is the number of ways to tile the 1xn board with triominoes (ie. size 1x3) and squares (ie. size 1x1) such that there are either none or one squares between triominoes, no squares at either end of the board, and there is at least one triomino. - _Enrique Navarrete_, Sep 07 2024
%H A017817 Seiichi Manyama, <a href="/A017817/b017817.txt">Table of n, a(n) for n = 0..10000</a>
%H A017817 Vladimir Baltic, <a href="https://doi.org/10.2298/AADM1000008B">On the number of certain types of strongly restricted permutations</a>, Applicable Analysis and Discrete Mathematics 4 (2010), 119-135.
%H A017817 Johann Cigler, <a href="https://arxiv.org/abs/2212.02118">Recurrences for certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials</a>, arXiv:2212.02118 [math.NT], 2022.
%H A017817 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=484">Encyclopedia of Combinatorial Structures 484</a>
%H A017817 Milan Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL19/Janjic/janjic73.html">Binomial Coefficients and Enumeration of Restricted Words</a>, Journal of Integer Sequences, 2016, Vol 19, #16.7.3.
%H A017817 E. Wilson, <a href="http://www.anaphoria.com/meruone.PDF">The Scales of Mt. Meru</a>
%H A017817 <a href="/index/Tu#2wis">Index entries for two-way infinite sequences</a>
%H A017817 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1,1).
%F A017817 G.f.: 1/(1-x^3-x^4).
%F A017817 a(n)/a(n-1) tends to A060007. - _Gary W. Adamson_, Oct 22 2006
%F A017817 a(n) = Sum_{k=0..floor(n/3)} binomial(k,n-3*k). - _Seiichi Manyama_, Mar 06 2019
%t A017817 LinearRecurrence[{0,0,1,1}, {1,0,0,1}, 60] (* _G. C. Greubel_, Mar 05 2019 *)
%o A017817 (PARI) a(n)=polcoeff(if(n<0,(1+x)/(1+x-x^4),1/(1-x^3-x^4)) +x*O(x^abs(n)), abs(n))
%o A017817 (Magma) I:=[1,0,0,1]; [n le 4 select I[n] else Self(n-3) +Self(n-4): n in [1..60]]; // _G. C. Greubel_, Mar 05 2019
%o A017817 (Sage) (1/(1-x^3-x^4)).series(x, 60).coefficients(x, sparse=False) # _G. C. Greubel_, Mar 05 2019
%o A017817 (GAP) a:=[1,0,0,1];; for n in [5..60] do a[n]:=a[n-3]+a[n-4]; od; a; # _G. C. Greubel_, Mar 05 2019
%Y A017817 A003269(n) = a(-4-n)(-1)^n.
%K A017817 nonn,easy
%O A017817 0,8
%A A017817 _N. J. A. Sloane_
%E A017817 More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Dec 17 1999