Search: id:A000100
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%I A000100 M1394 N0543
%S A000100 0,0,0,1,2,5,11,23,47,94,185,360,694,1328,2526,4781,9012,16929,31709,59247,
%T A000100 110469,205606,382087,709108,1314512,2434364,4504352,8328253,15388362,
%U A000100 28417385,52451811,96771787,178473023,329042890,606466009,1117506500
%N A000100 a(n) = number of compositions of n in which the maximum part size is 
               3.
%C A000100 For n > 5, a(n) - (a(n-3)+a(n-2)+a(n-1)) = F(n-2) where F(i) is the i-th 
               Fibonacci number; e.g. 11 - (1+2+5) = 3, 23 - (2+5+11) = 8; also 
               lim n->inf a(n)/(a(n-1)+a(n-2)+a(n-3)) = 1 and lim n->inf a(n)a(n-2)/
               a(n-1)^2 = 1 - Gerald McGarvey (Gerald.McGarvey(AT)comcast.net), 
               Jun 26 2004
%C A000100 a(n) is also the number of binary sequences of length n-1 in which the 
               longest run of 0's is exactly two. [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), 
               Nov 06 2008]
%D A000100 A. T. Benjamin and J. J. Quinn, Proofs that really count: the art of 
               combinatorial proof, M.A.A. 2003, p. 47, ex. 4.
%D A000100 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 
               155.
%D A000100 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000100 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000100 J. L. Yucas, Counting special sets of binary Lyndon words, Ars Combin., 
               31 (1991), 21-29.
%H A000100 T. D. Noe, <a href="b000100.txt">Table of n, a(n) for n=0..200</a>
%H A000100 Nick Hobson, <a href="a000100.py.txt">Python program for this sequence</
               a>
%F A000100 G.f.: x^3/((1-x-x^2)*(1-x-x^2-x^3)).
%F A000100 a(n+3) = Sum[k=0..n, F(k)*T(n-k) ], F(i)=A000045(i+1), T(i)=A000073(i+2).
%F A000100 a(n)=2*a(n-1)+a(n-2)-a(n-3)-2*a(n-4)-a(n-5). Convolution of Fibonacci 
               and Tribonacci numbers (A000045 and A000073). - Frank Adams-Watters 
               (FrankTAW(AT)Netscape.net), Jan 13 2006
%e A000100 For example, a(5)=5 counts 1+1+3, 2+3, 3+2, 3+1+1, 1+3+1. - David Callan 
               (callan(AT)stat.wisc.edu), Dec 09 2004
%e A000100 a(5)=5 because there are 5 binary sequences of length 4 in which the 
               longest run of consecutive 0's is exactly two. 0010,0011,0100,1001,
               1100 [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Nov 
               06 2008]
%p A000100 (Maple) a := n -> (Matrix(5, (i,j)-> if (i=j-1) then 1 elif j=1 then 
               [2,1,-1,-2,-1][i] else 0 fi)^(n))[1,4] ; seq (a(n), n=0..35); [From 
               Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 04 2008]
%Y A000100 Cf. A000045.
%Y A000100 Sequence in context: A140992 A093053 A075712 this_sequence A083005 A133489 
               A060153
%Y A000100 Adjacent sequences: A000097 A000098 A000099 this_sequence A000101 A000102 
               A000103
%K A000100 nonn,easy,nice
%O A000100 0,5
%A A000100 N. J. A. Sloane (njas(AT)research.att.com).
%E A000100 More terms from Henry Bottomley (se16(AT)btinternet.com), Dec 15 2000
%E A000100 Better definition from David Callan and Frank Adams-Watters.

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