

A342225


Total number of ordered graceful labelings of graphs with n edges.


2



1, 2, 4, 12, 40, 182, 906, 5404, 35494, 264178, 2124078, 18965372, 181080940, 1879988162, 20764521072, 246377199752, 3085635516364, 41182472709986, 577129788232678, 8552244962978250, 132591961730782524, 2161198867136837458
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OFFSET

1,2


COMMENTS

Also the number of sequences l_0, l_1, ..., l_{n1} such that 0 <= l_k <= k and such that l_j+nj != l_k for 0 <= j,k < n.
Ordered graceful labelings were originally called "near alphalabelings". They have also been called "gracious labelings" and "beta^+labelings.
The corresponding number of "true" alphalabelings is A005193(n).
The corresponding number of unrestricted graceful labelings is A000142(n).
The corresponding number of unrestricted graceful labelings of bipartite graphs is 2*A334613(n+1).


REFERENCES

D. E. Knuth, The Art of Computer Programming, Volume 4B, Section 7.2.2.3 will have an exercise based on this sequence.


LINKS



EXAMPLE

For n=4 the a(4)=12 solutions l_0l_1l_2l_3 are 0000, 0001, 0011, 0012, 0020, 0022, 0101, 0103, 0111, 0112, 0122, 0123. (Of these, 0022 and 0103 are not counted by A005193.)


CROSSREFS



KEYWORD

nonn,more


AUTHOR



EXTENSIONS



STATUS

approved



