%I A000102 M1409 N0551
%S A000102 0,0,0,0,1,2,5,12,27,59,127,269,563,1167,2400,4903,9960,20135,40534,81300,
%T A000102 162538,324020,644282,1278152,2530407,5000178,9863763,19427976,
%U A000102 38211861,75059535,147263905,288609341,565047233,1105229439,2159947998
%N A000102 a(n) = number of compositions of n in which the maximum part size is 
               4.
%C A000102 a(n) is also the number of binary sequences of length n-1 in which the 
               longest run of consecutive 0's is exactly three. [From Geoffrey Critzer 
               (critzer.geoffrey(AT)usd443.org), Nov 06 2008]
%D A000102 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 
               155.
%D A000102 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000102 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000102 J. L. Yucas, Counting special sets of binary Lyndon words, Ars Combin., 
               31 (1991), 21-29.
%H A000102 T. D. Noe, <a href="b000102.txt">Table of n, a(n) for n=0..200</a>
%H A000102 Nick Hobson, <a href="a000102.py.txt">Python program for this sequence</
               a>
%F A000102 G.f.: x^4/(1-x-x^2-x^3)/(1-x-x^2-x^3-x^4).
%F A000102 a(n)=2*a(n-1)+a(n-2)-2*a(n-4)-3*a(n-5)-2*a(n-6)-a(n-7). Convolution of 
               Tribonacci and Tetranacci numbers (A000073 and A000078). - Frank 
               Adams-Watters (FrankTAW(AT)Netscape.net), Jan 13 2006
%e A000102 For example, a(6)=5 counts 1+1+4, 2+4, 4+2, 4+1+1, 1+4+1. - David Callan 
               (callan(AT)stat.wisc.edu), Dec 09 2004
%e A000102 a(6)=5 because there are 5 binary sequences of length 5 in which the 
               longest run of consecutive 0's is exactly 3; 00010,00011,01000,10001,
               11000 [From Geoffrey Critzer (critzer.geoffrey(AT)usd443.org), Nov 
               06 2008]
%p A000102 a:= n-> (Matrix(7, (i,j)-> if i+1=j then 1 elif j=1 then [2, 1, 0, -2, 
               -3, -2, -1][i] else 0 fi)^n)[1,5]: seq (a(n), n=0..40); [From Alois 
               P. Heinz (heinz(AT)hs-heilbronn.de), Oct 07 2008]
%Y A000102 Sequence in context: A129983 A083378 A116712 this_sequence A086589 A091596 
               A077863
%Y A000102 Adjacent sequences: A000099 A000100 A000101 this_sequence A000103 A000104 
               A000105
%K A000102 nonn,nice,easy
%O A000102 0,6
%A A000102 N. J. A. Sloane (njas(AT)research.att.com).
%E A000102 More terms from Sascha Kurz (sascha.kurz(AT)uni-bayreuth.de), Aug 15 
               2002
%E A000102 Definition improved by David Callan and Frank Adams-Watters.