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%I #49 Jul 27 2022 10:30:42
%S 1,0,1,0,2,0,3,0,5,0,8,0,13,0,21,0,34,0,55,0,89,0,144,0,233,0,377,0,
%T 610,0,987,0,1597,0,2584,0,4181,0,6765,0,10946,0,17711,0,28657,0,
%U 46368,0,75025,0,121393,0,196418,0,317811,0,514229,0,832040,0,1346269
%N Fibonacci numbers interspersed with zeros.
%C 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={0,2}.
%C Number of compositions of n into elements of the set {2,4}.
%C a(n-2) is the number of circular arrangements of the first n positive integers such that adjacent terms have absolute difference 1 or 3. - _Ethan Patrick White_, Jun 24 2020
%D D. H. Lehmer, Permutations with strongly restricted displacements. Combinatorial theory and its applications, II (Proc. Colloq., Balatonfured, 1969), pp. 755-770. North-Holland, Amsterdam, 1970.
%H G. C. Greubel, <a href="/A079977/b079977.txt">Table of n, a(n) for n = 0..1000</a>
%H Vladimir Baltic, <a href="http://pefmath.etf.rs/vol4num1/AADM-Vol4-No1-119-135.pdf">On the number of certain types of strongly restricted permutations</a>, Applicable Analysis and Discrete Mathematics 4 (2010), 119-135.
%H Ethan P. White, Richard K. Guy, and Renate Scheidler, <a href="https://arxiv.org/abs/2006.15250">Difference Necklaces</a>, arXiv:2006.15250 [math.CO], 2020.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (0,1,0,1).
%F a(n) = A000045(k+1) if n=2k, a(n)=0 otherwise.
%F a(n) = a(n-2) + a(n-4).
%F G.f.: 1/(1 - x^2 - x^4).
%t Riffle[Fibonacci[Range[50]],0] (* _Harvey P. Dale_, Dec 20 2015 *)
%o (PARI) a(n)=if(n%2,0,fibonacci(n/2+1)) \\ _Charles R Greathouse IV_, Jun 11 2015
%o (Magma)
%o A079977:= func< n | (1+(-1)^n)*Fibonacci(Floor((n+2)/2))/2 >;
%o [A079977(n): n in [0..50]]; // _G. C. Greubel_, Jul 25 2022
%o (SageMath)
%o def A079977(n): return ((n+1)%2)*fibonacci((n+2)//2)
%o [A079977(n) for n in (0..50)] # _G. C. Greubel_, Jul 25 2022
%Y Cf. A000045, A002524, A002525, A002526, A002527, A002528, A002529, A072827.
%Y Cf. A072850, A072851, A072852, A072853, A072854, A072855, A072856, A099574.
%Y Cf. A079955 - A080014.
%K nonn,easy
%O 0,5
%A _Vladimir Baltic_, Feb 17 2003
%E Editorial note: normally the alternate zeros are omitted from sequences like this. This entry is an exception. - _N. J. A. Sloane_
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