A079962 - OEIS

%I #44 Jan 22 2022 20:58:10

%S 1,1,1,2,3,5,9,14,22,36,58,94,153,247,399,646,1045,1691,2737,4428,

%T 7164,11592,18756,30348,49105,79453,128557,208010,336567,544577,

%U 881145,1425722,2306866,3732588,6039454,9772042,15811497,25583539,41395035

%N 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}.

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

%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="/A079962/b079962.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 Vol. 4, No 1 (2010), 119-135

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,0,1,1).

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

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

%F a(n+1)/a(n) -> golden ratio A001622. - _Roger L. Bagula_, Mar 13 2006

%F a(n) + a(n+2) + a(n+4) = Fibonacci(n+5). - _Mark Dols_, Aug 20 2010

%F a(n) = round(Fibonacci(n+3)/4). - _Mircea Merca_, Jan 04 2011

%F a(n+6) - a(n) = A000045(n+6). - _Paul Curtz_, Jun 29 2013

%F a(n) + a(n+1) + a(n+2) = A024490(n+6). - _R. J. Mathar_, Jun 30 2013

%F a(n) - a(n-1) + a(n-2) = A094686(n). - _R. J. Mathar_, Jun 30 2013

%F 4*a(n) = A057078(n) + A010892(n) + A000045(n+3). - _R. J. Mathar_, Nov 02 2016

%p with(combinat,fibonacci): seq(round(fibonacci(n+3)/4),n=0..38) # _Mircea Merca_, Jan 04 2011

%t LinearRecurrence[{1,0,1,0,1,1}, {1,1,1,2,3,5}, 41] (* _G. C. Greubel_, Jan 21 2022 *)

%o (PARI) a(n)=fibonacci(n+3)\/4 \\ _Charles R Greathouse IV_, Oct 07 2015

%o (Magma) [Round(Fibonacci(n+3)/4): n in [0..40]]; // _G. C. Greubel_, Jan 21 2022

%o (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

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

%K nonn,easy

%O 0,4

%A _Vladimir Baltic_, Feb 19 2003