A079962 Number of permutations satisfying -k <= p(i) - i <= r and p(i) - i not in I, i=1..n, with k=1, r=5, I={1,3}.

1, 1, 1, 2, 3, 5, 9, 14, 22, 36, 58, 94, 153, 247, 399, 646, 1045, 1691, 2737, 4428, 7164, 11592, 18756, 30348, 49105, 79453, 128557, 208010, 336567, 544577, 881145, 1425722, 2306866, 3732588, 6039454, 9772042, 15811497, 25583539, 41395035

Offset: 0

Author

Vladimir Baltic, Feb 19 2003

Keywords

• nonn
• easy

Comments

Number of compositions (ordered partitions) of n into elements of the set {1,3,5,6}. - Mark Dols, Aug 20 2010

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 Vol. 4, No 1 (2010), 119-135
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,0,1,1).

Crossrefs

Cf. A002524, A002525, A002526, A002527, A002528, A002529, A072827, A072850, A072851, A072852, A072853, A072854, A072855, A072856, A079955 - A080014.

Programs

Magma

[Round(Fibonacci(n+3)/4): n in [0..40]]; // G. C. Greubel, Jan 21 2022

Maple

with(combinat,fibonacci): seq(round(fibonacci(n+3)/4),n=0..38) # Mircea Merca, Jan 04 2011

Mathematica

LinearRecurrence[{1,0,1,0,1,1}, {1,1,1,2,3,5}, 41] (* G. C. Greubel, Jan 21 2022 *)

PARI

a(n)=fibonacci(n+3)\/4 \\ Charles R Greathouse IV, Oct 07 2015

Sage

[(1/4)*(fibonacci(n+3) + chebyshev_U(n,1/2) + chebyshev_U(2*n,1/2)) for n in (0..40)] # G. C. Greubel, Jan 21 2022

Formula

a(n) = a(n-1) + a(n-3) + a(n-5) + a(n-6).

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

a(n+1)/a(n) -> golden ratio A001622. - Roger L. Bagula, Mar 13 2006

a(n) + a(n+2) + a(n+4) = Fibonacci(n+5). - Mark Dols, Aug 20 2010

a(n) = round(Fibonacci(n+3)/4). - Mircea Merca, Jan 04 2011

a(n+6) - a(n) = A000045(n+6). - Paul Curtz, Jun 29 2013

a(n) + a(n+1) + a(n+2) = A024490(n+6). - R. J. Mathar, Jun 30 2013

a(n) - a(n-1) + a(n-2) = A094686(n). - R. J. Mathar, Jun 30 2013

4*a(n) = A057078(n) + A010892(n) + A000045(n+3). - R. J. Mathar, Nov 02 2016