%I A005314 M0709
%S A005314 0,1,2,3,5,9,16,28,49,86,151,265,465,816,1432,2513,4410,7739,13581,
%T A005314 23833,41824,73396,128801,226030,396655,696081,1221537,2143648,3761840,
%U A005314 6601569,11584946,20330163,35676949,62608681,109870576,192809420
%N A005314 For n = 0, 1, 2, a(n) = n; thereafter, a(n) = 2a(n-1)-a(n-2)+a(n-3).
%C A005314 Number of compositions of n into parts congruent to {1,2} mod 4. - Vladeta 
               Jovovic (vladeta(AT)eunet.rs), Mar 10 2005
%C A005314 a(n)/a(n-1) tends to 1.75487766625...; an eigenvalue of the matrix M 
               and a root to the characteristic polynomial. - Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               May 25 2007
%C A005314 Starting with offset 1 = INVERT transform of (1, 1, 0, 0, 1, 1, 0, 0,
               ...). [From Gary W. Adamson (qntmpkt(AT)yahoo.com), May 04 2009]
%D A005314 R. L. Graham and N. J. A. Sloane, Anti-Hadamard matrices, Linear Alg. 
               Applic., 62 (1984), 113-137.
%D A005314 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A005314 T. D. Noe, <a href="b005314.txt">Table of n, a(n) for n=0..400</a>
%H A005314 P. Chinn and S. Heubach, <a href="http://www.cs.uwaterloo.ca/journals/
               JIS/index.html">Integer Sequences Related to Compositions without 
               2's</a>, J. Integer Seqs., Vol. 6, 2003.
%H A005314 INRIA Algorithms Project, <a href="http://algo.inria.fr/bin/encyclopedia?Search=ECSnb&argsearch=426">
               Encyclopedia of Combinatorial Structures 426</a>
%H A005314 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/MasterThesis.pdf">
               Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures</
               a>, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 
               1992.
%H A005314 S. Plouffe, <a href="http://www.lacim.uqam.ca/%7Eplouffe/articles/FonctionsGeneratrices.pdf">
               1031 Generating Functions and Conjectures</a>, Universit\'{e} du 
               Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
%F A005314 G.f.: x/(1-2*x+x^2-x^3). a(n) = Sum_{k=0..[(2n-1)/3]} binomial(n-1-[k/
               2], k), where [x]=floor(x). - Paul D. Hanna (pauldhanna(AT)juno.com), 
               Oct 22 2004
%F A005314 a(n) = Sum [k=0..n, C(n-k, 2k+1) ].
%F A005314 23*a_n = 3*P_{2n+2} + 7*P_{2n+1} - 2*P_{2n}, where P_n are the Perrin 
               numbers, A001608. - D. E. Knuth, Dec 09 2008
%F A005314 G.f. (z-1)*(1+z**2)/(-1+2*z+z**3-z**2) for the augmented version 1, 1, 
               2, 3, 5, 9, 16, 28, 49, 86, 151,... was given in S. Plouffe's thesis 
               of 1992.
%F A005314 a(n) = a(n-1)+a(n-2)+a(n-4) = a(n-2)+A049853(n-1) = a(n-1)+A005251(n) 
               = sum_{i <= n} (A005251(i)).
%F A005314 a(n) = Sum(binomial(n-k, 2k+1), {k=0...(n-1)/3}) - Richard Ollerton (r.ollerton(AT)uws.edu.au), 
               May 12 2004
%F A005314 M^n*[1,0,0] = [a(n-2), a(n-1), a]; where M = the 3 X 3 matrix [0,1,0; 
               0,0,1; 1,-1,2]. Example M^5*[1,0,0] = [3,5,9]. - Gary W. Adamson 
               (qntmpkt(AT)yahoo.com), May 25 2007
%o A005314 (PARI) a(n)=sum(k=0,(2*n-1)\3,binomial(n-1-k\2,k))
%Y A005314 Equals row sums of triangle A099557. - Paul D. Hanna (pauldhanna(AT)juno.com), 
               Oct 22 2004
%Y A005314 Cf. A099557, A005251.
%Y A005314 Sequence in context: A134009 A018160 A079960 this_sequence A099529 A088352 
               A002572
%Y A005314 Adjacent sequences: A005311 A005312 A005313 this_sequence A005315 A005316 
               A005317
%K A005314 nonn
%O A005314 0,3
%A A005314 N. J. A. Sloane (njas(AT)research.att.com).
%E A005314 More terms and additional formulae from Henry Bottomley (se16(AT)btinternet.com), 
               Jul 21 2000
%E A005314 Plouffe's g.f. edited by R. J. Mathar, May 12 2008