%I A005045 M2536
%S A005045 0,0,1,3,6,10,17,25,37,51,70,92,121,153,194,240,296,358,433,515,612,718,
%T A005045 841,975,1129,1295,1484,1688,1917,2163,2438,2732,3058,3406,3789,
%U A005045 4197,4644,5118,5635,6183,6777,7405,8084,8800,9571,10383,11254
%N A005045 Number of restricted 3 X 3 matrices with row and column sums n.
%C A005045 More precisely, consider 3 X 3 matrices with entries chosen from {0, 
               1, ..., n-1}, in which each row and column sums to n, where n >= 
               2. Then a(n) is the number of equivalence classes of such matrices 
               under permutions of rows and columns and transpositions.
%D A005045 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A005045 E. J. Morgan, Construction of Block Designs and Related Results. Ph.D. 
               Dissertation, Univ. Queensland, 1978.
%D A005045 E. J. Morgan, On 3 X 3 matrices with constant row and column sum, Abstract 
               763-05-13, Notices Amer. Math. Soc., 26 (1979), page A-27.
%H A005045 M. F. Hasler, <a href="b005045.txt">Table of n, a(n) for n=0,...,1000</
               a>.
%H A005045 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 A005045 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.
%H A005045 R. J. Mathar, <a href="a005045.pdf">OEIS A005045</a> [Proof of g.f.. 
               for 3 of the 12 cases]
%F A005045 Let n = 3k, 3k-1 or 3k-2 according as n == 0, 2 or 1 mod 3, for n>=3. 
               Then a(n) = Sum_{ i=1,...,n-k } Sum_{ m=max(0,2i-n),...,floor(i/2) 
               } Sum_{ r=0,...,floor(i/2)-m } c(i,m,r), where c(i,m,r) = n-2i+m+1 
               when m+r != i/2, or = floor((n-2i+m+2)/2) when m+r = i/2. [Typos 
               corrected by Peter Pein, May 13 2008]
%F A005045 G.f.: -x**2*(-x**5+x**6-x**3+x+1)/((x**2+1)*(x**2+x+1)*(x+1)**2*(x-1)**5). 
               This was conjectured by S. Plouffe in his 1992 dissertation and is 
               now known to be correct, although it may be that all the details 
               of the proof have not been written down. See the Mathar link for 
               details.
%e A005045 a(2) = 1:
%e A005045 110
%e A005045 101
%e A005045 011
%e A005045 a(3) = 3:
%e A005045 111 210 210
%e A005045 111 102 111
%e A005045 111 021 012
%p A005045 A005045:=-z**2*(-z**5+z**6-z**3+z+1)/((z**2+1)*(z**2+z+1)*(z+1)**2*(z-1)**5); 
               [Conjectured by S. Plouffe in his 1992 dissertation. See formula 
               lines here for the proof of correctness.]
%t A005045 Block[{k = Floor[(n + 2)/3]}, Sum[Sum[Sum[If[m + r == i/2, Floor[(n - 
               2*i + m + 2)/2], n - 2*i + m + 1], {r, 0, Floor[i/2 - m]}], {m, Max[2*i 
               - n, 0], Floor[i/2]}], {i, 1, n - k}]]; Table[an, {n, 2, 100}] (from 
               Peter Pein, May 13 2008)
%o A005045 (PARI from M. F. Hasler, Version 1, May 13 2008:)
%o A005045 A005045(n)={sum( i=1,n-(n+2)\3, sum( m=max(0,2*i-n),i\2, sum( r=0,i\2-m, 
               if( m+r!=i/2, n-2*i+m+1, (n-2*i+m+2)\2))))}
%o A005045 (PARI from M. F. Hasler, Version 2, much faster, May 13 2008:)
%o A005045 A005045(n)={sum( i=1,(2*n)\3, sum( m=max(0,2*i-n),i\2, (n-2*i+m+1)*((i+1)\2-m)+(i%2==0)*(n-2*i+m+2)\2))}
%Y A005045 Cf. A002817 for another version.
%Y A005045 Sequence in context: A038505 A119971 A094272 this_sequence A069241 A092263 
               A076251
%Y A005045 Adjacent sequences: A005042 A005043 A005044 this_sequence A005046 A005047 
               A005048
%K A005045 nonn,nice
%O A005045 0,4
%A A005045 N. J. A. Sloane (njas(AT)research.att.com).
%E A005045 Edited by N. J. A. Sloane (njas(AT)research.att.com), May 12 2008, May 
               13 2008
%E A005045 More terms from Peter Pein, May 13 2008