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A358244
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Number of n-regular, N_0-weighted pseudographs on 2 vertices with total edge weight 4, up to isomorphism.
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7
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1, 6, 13, 27, 38, 55, 67, 85, 97, 115, 127, 145, 157, 175, 187, 205, 217, 235, 247, 265, 277, 295, 307, 325, 337, 355, 367, 385, 397, 415, 427, 445, 457, 475, 487, 505, 517, 535, 547, 565, 577, 595, 607, 625, 637, 655, 667, 685, 697, 715, 727, 745, 757, 775
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OFFSET
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1,2
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COMMENTS
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Pseudographs are finite graphs with undirected edges without identity, where parallel edges between the same vertices and loops are allowed.
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LINKS
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FORMULA
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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
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EXAMPLE
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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.
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PROG
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(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])
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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