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A005045
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Number of restricted 3 X 3 matrices with row and column sums n.
(Formerly M2536)
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2
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0, 0, 1, 3, 6, 10, 17, 25, 37, 51, 70, 92, 121, 153, 194, 240, 296, 358, 433, 515, 612, 718, 841, 975, 1129, 1295, 1484, 1688, 1917, 2163, 2438, 2732, 3058, 3406, 3789, 4197, 4644, 5118, 5635, 6183, 6777, 7405, 8084, 8800, 9571, 10383, 11254
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| 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.
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REFERENCES
| E. J. Morgan, Construction of Block Designs and Related Results. Ph.D. Dissertation, Univ. Queensland, 1978.
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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| M. F. Hasler, Table of n, a(n) for n=0,...,1000.
R. J. Mathar, OEIS A005045 [Proof of g.f.. for 3 of the 12 cases]
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
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FORMULA
| 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]
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.
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EXAMPLE
| a(2) = 1:
110
101
011
a(3) = 3:
111 210 210
111 102 111
111 021 012
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MAPLE
| 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.]
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MATHEMATICA
| 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)
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PROG
| (PARI from M. F. Hasler, Version 1, May 13 2008:)
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))))}
(PARI from M. F. Hasler, Version 2, much faster, May 13 2008:)
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))}
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CROSSREFS
| Cf. A002817 for another version.
Sequence in context: A038505 A119971 A094272 * A189376 A069241 A092263
Adjacent sequences: A005042 A005043 A005044 * A005046 A005047 A005048
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KEYWORD
| nonn,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com), May 12 2008, May 13 2008
More terms from Peter Pein, May 13 2008
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