

A005045


Number of restricted 3 X 3 matrices with row and column sums n.
(Formerly M2536)


4



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
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OFFSET

0,4


COMMENTS

More precisely, consider 3 X 3 matrices with entries chosen from {0, 1, ..., n1}, 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.


REFERENCES

E. J. Morgan, On 3 X 3 matrices with constant row and column sum, Abstract 7630513, Notices Amer. Math. Soc., 26 (1979), page A27.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


LINKS

M. F. Hasler, Table of n, a(n) for n = 0..1000
E. J. Billington (née Morgan) and N. J. A. Sloane, Correspondence, 19781991.
P. Lisonek, Quasipolynomials: A case study in experimental combinatorics, RISCLinz Report Series No. 9318, 1983. (Annotated scanned copy)
R. J. Mathar, OEIS A005045 [Proof of g.f. for 3 of the 12 cases]
E. J. Morgan, Construction of Block Designs and Related Results, Ph.D. Dissertation, Univ. Queensland, 1978; Bull. Austral. Math. Soc., Volume 19, Issue 1 August 1978, pp. 139140.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
Index entries for linear recurrences with constant coefficients, signature (2,0,1,0,2,2,0,1,0,2,1).


FORMULA

Let n = 3k, 3k1 or 3k2 according as n == 0, 2 or 1 mod 3, for n >= 3. Then a(n) = Sum_{i=1..nk} Sum_{m=max(0,2in)..floor(i/2)} Sum_{r=0..floor(i/2)m} c(i,m,r), where c(i,m,r) = n2i+m+1 when m+r != i/2, or = floor((n2i+m+2)/2) when m+r = i/2. [Typos corrected by Peter Pein, May 13 2008]
G.f.: x^2*(x^5+x^6x^3+x+1)/((x^2+1)*(x^2+x+1)*(x+1)^2*(x1)^5). This was conjectured by Simon 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.


EXAMPLE

a(2) = 1:
110
101
011
a(3) = 3:
111 210 210
111 102 111
111 021 012


MAPLE

A005045:=z**2*(z**5+z**6z**3+z+1)/((z**2+1)*(z**2+z+1)*(z+1)**2*(z1)**5); # conjectured by Simon Plouffe in his 1992 dissertation; see formula lines here for the proof of correctness


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)
LinearRecurrence[{2, 0, 1, 0, 2, 2, 0, 1, 0, 2, 1}, {0, 0, 1, 3, 6, 10, 17, 25, 37, 51, 70}, 50] (* Harvey P. Dale, Nov 15 2018 *)


PROG

(PARI)
A005045(n)={sum( i=1, n(n+2)\3, sum( m=max(0, 2*in), i\2, sum( r=0, i\2m, if( m+r!=i/2, n2*i+m+1, (n2*i+m+2)\2))))} \\ M. F. Hasler, Version 1, May 13 2008
(PARI)
A005045(n)={sum( i=1, (2*n)\3, sum( m=max(0, 2*in), i\2, (n2*i+m+1)*((i+1)\2m)+(i%2==0)*(n2*i+m+2)\2))} \\ M. F. Hasler, Version 2, much faster, May 13 2008
(PARI) concat(vector(2), Vec(x^2*(1 + x  x^3  x^5 + x^6) / ((1  x)^5*(1 + x)^2*(1 + x^2)*(1 + x + x^2)) + O(x^60))) \\ Colin Barker, Apr 22 2017


CROSSREFS

Cf. A002817 for another version.
Sequence in context: A236326 A308699 A286304 * A189376 A069241 A092263
Adjacent sequences: A005042 A005043 A005044 * A005046 A005047 A005048


KEYWORD

nonn,nice,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

Edited by N. J. A. Sloane, May 12 2008, May 13 2008
More terms from Peter Pein, May 13 2008


STATUS

approved



