A079977 Fibonacci numbers interspersed with zeros.

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, 610, 0, 987, 0, 1597, 0, 2584, 0, 4181, 0, 6765, 0, 10946, 0, 17711, 0, 28657, 0, 46368, 0, 75025, 0, 121393, 0, 196418, 0, 317811, 0, 514229, 0, 832040, 0, 1346269

Offset: 0

Author

Vladimir Baltic, Feb 17 2003

Keywords

• nonn
• easy

Comments

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}.

Number of compositions of n into elements of the set {2,4}.

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

References

• 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.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Vladimir Baltic, On the number of certain types of strongly restricted permutations, Applicable Analysis and Discrete Mathematics 4 (2010), 119-135.
• Ethan P. White, Richard K. Guy, and Renate Scheidler, Difference Necklaces, arXiv:2006.15250 [math.CO], 2020.
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,1).

Crossrefs

Cf. A000045, A002524, A002525, A002526, A002527, A002528, A002529, A072827.

Cf. A072850, A072851, A072852, A072853, A072854, A072855, A072856, A099574.

Cf. A079955 - A080014.

Programs

Magma

A079977:= func< n | (1+(-1)^n)*Fibonacci(Floor((n+2)/2))/2 >;
[A079977(n): n in [0..50]]; // G. C. Greubel, Jul 25 2022

Mathematica

Riffle[Fibonacci[Range[50]],0] (* Harvey P. Dale, Dec 20 2015 *)

PARI

a(n)=if(n%2,0,fibonacci(n/2+1)) \\ Charles R Greathouse IV, Jun 11 2015

SageMath

def A079977(n): return ((n+1)%2)*fibonacci((n+2)//2)
[A079977(n) for n in (0..50)] # G. C. Greubel, Jul 25 2022

Formula

a(n) = A000045(k+1) if n=2k, a(n)=0 otherwise.

a(n) = a(n-2) + a(n-4).

G.f.: 1/(1 - x^2 - x^4).

Extensions

Editorial note: normally the alternate zeros are omitted from sequences like this. This entry is an exception. - N. J. A. Sloane