A336529 a(n) = (n^3+5*n+3)/3 + 2*floor(n/2) + a(n-2), with a(0)=1 and a(1)=3.

1, 3, 10, 20, 43, 75, 132, 208, 325, 475, 686, 948, 1295, 1715, 2248, 2880, 3657, 4563, 5650, 6900, 8371, 10043, 11980, 14160, 16653, 19435, 22582, 26068, 29975, 34275, 39056, 44288, 50065, 56355, 63258, 70740, 78907, 87723, 97300, 107600, 118741, 130683, 143550, 157300

Offset: 0

Author

Yu Hin Au, Jul 24 2020

Keywords

• nonn
• easy

Comments

Let S be a fixed matching of size 2 in a complete n-uniform hypergraph G with >= 4n vertices. Given T,T' (each also a matching of size 2), define the equivalence relation where T ~ T' if and only if there exists an automorphism of G that maps every hyperedge in T to a hyperedge in T' while mapping every hyperedge in S to a hyperedge in S. Then the number of equivalence classes is a(n).

a(n) is the number of equivalence classes of 2 X 2 matrices with nonnegative integer entries where each row and column sum to at most n, such that two matrices are related if one can be obtained from the other by permuting rows and columns.

Examples

			To see a(2)=10, let S = {{1,2},{3,4}}. Then a representative from each of the 10 equivalence classes are
  1. {{1,2}, {3,4}}
  2. {{1,3}, {2,4}}
  3. {{1,5}, {3,4}}
  4. {{1,3}, {4,5}}
  5. {{1,2}, {5,6}}
  6. {{1,3}, {5,6}}
  7. {{1,5}, {2,6}}
  8. {{1,5}, {3,6}}
  9. {{1,5}, {6,7}}
  10. {{5,6}, {7,8}}
Likewise, in the 2 X 2 matrix interpretation, a representative from each of the a(2)=10 equivalence classes are
  [2 0 ; 0 2]
  [1 1 ; 1 1]
  [2 0 ; 0 1]
  [1 1 ; 1 0]
  [2 0 ; 0 0]
  [1 1 ; 0 0]
  [1 0 ; 1 0]
  [1 0 ; 0 1]
  [1 0 ; 0 0]
  [0 0 ; 0 0]
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..10000
• Yu Hin Au, Nathan Lindzey, and Levent Tunçel, Matchings, hypergraphs, association schemes, and semidefinite optimization, arXiv:2008.08628 [math.CO], 2020.
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Crossrefs

Cf. A316587.

Programs

Mathematica

Nest[Append[#1, (#2^3 + 5 #2 + 3)/3 + 2*Floor[#2/2] + #1[[-2]] ] & @@ {#, Length@ #} &, {1, 3}, 42] (* Michael De Vlieger, Nov 04 2020 *)
LinearRecurrence[{3,-1,-5,5,1,-3,1},{1,3,10,20,43,75,132},60] (* Harvey P. Dale, May 28 2021 *)

PARI

Vec((1 + 2*x^2 - 2*x^3 + 3*x^4) / ((1 - x)^5*(1 + x)^2) + O(x^40)) \\ Colin Barker, Nov 05 2020

Formula

a(n) = 3*a(n-1) - a(n-2) - 5*a(n-3) + 5*a(n-4) + a(n-5) - 3*a(n-6) + a(n-7). - Wesley Ivan Hurt, Nov 04 2020

G.f.: (1 + 2*x^2 - 2*x^3 + 3*x^4) / ((1 - x)^5*(1 + x)^2). - Colin Barker, Nov 05 2020