OFFSET
1,2
COMMENTS
Pseudographs are finite graphs with undirected edges without identity, where parallel edges between the same vertices and loops are allowed.
LINKS
Lars Göttgens, Table of n, a(n) for n = 1..10000
J. Flake and V. Mackscheidt, Interpolating PBW Deformations for the Orthosymplectic Groups, arXiv:2206.08226 [math.RT], 2022.
Eric Weisstein's World of Mathematics, Pseudograph.
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).
FORMULA
Apparently a(n) = 6*A047209(n-2) + 1 for n >= 6, i.e., terms satisfy the linear recurrence a(n) = a(n-1) + a(n-2) - a(n-3) for n >= 9. - Hugo Pfoertner, Dec 02 2022
From Andrew Howroyd, Nov 23 2025: (Start)
The above observation is correct. See A390167 for an explanation.
a(n) = (30*n + 3*(-1)^n - 73)/2 for n >= 6.
G.f.: x*(1 + 5*x + 6*x^2 + 9*x^3 + 4*x^4 + 3*x^5 + x^6 + x^7)/((1 - x)^2*(1 + x)). (End)
EXAMPLE
For n = 2 the a(2) = 6 such pseudographs are:
1. two vertices connected by a 4-edge and a 0-edge,
2. two vertices connected by a 3-edge and a 1-edge,
3. two vertices connected by two 2-edges,
4. two vertices where one has a 4-loop and the other one has a 0-loop,
5. two vertices where one has a 3-loop and the other one has a 1-loop,
6. two vertices with a 2-loop each.
PROG
(Julia)
using Combinatorics
function A(n::Int)
sum_total = 4
result = 0
for num_loops in 0:div(n, 2)
num_cross = n - 2 * num_loops
for sum_cross in 0:sum_total
for sum_loop1 in 0:sum_total-sum_cross
sum_loop2 = sum_total - sum_cross - sum_loop1
if sum_loop2 == sum_loop1
result +=
div(
npartitions_with_zero(sum_loop2, num_loops) *
(npartitions_with_zero(sum_loop2, num_loops) + 1),
2,
) * npartitions_with_zero(sum_cross, num_cross)
elseif sum_loop2 > sum_loop1
result +=
npartitions_with_zero(sum_loop2, num_loops) *
npartitions_with_zero(sum_loop1, num_loops) *
npartitions_with_zero(sum_cross, num_cross)
end
end
end
end
return result
end
function npartitions_with_zero(n::Int, m::Int)
if m == 0
if n == 0
return 1
else
return 0
end
else
return Combinatorics.npartitions(n + m, m)
end
end
print([A(n) for n in 1:54])
(PARI) Vec((1 + 5*x + 6*x^2 + 9*x^3 + 4*x^4 + 3*x^5 + x^6 + x^7)/((1 - x)^2*(1 + x)) + O(x^50)) \\ Andrew Howroyd, Nov 23 2025
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Lars Göttgens, Nov 04 2022
STATUS
approved