OFFSET
1,2
COMMENTS
Also the number of sequences l_0, l_1, ..., l_{n-1} such that 0 <= l_k <= k and such that l_j+n-j != l_k for 0 <= j,k < n.
Ordered graceful labelings were originally called "near alpha-labelings". They have also been called "gracious labelings" and "beta^+-labelings.
The corresponding number of "true" alpha-labelings 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
S. I. El-Zanati, M. J. Kenig, and C. Vanden Eynden, Near α-labelings of bipartite graphs, Australasian Journal of Combinatorics, 21 (2000), 275-285.
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
Don Knuth, Mar 06 2021
EXTENSIONS
a(18)-a(22) from Bert Dobbelaere, Mar 09 2021
STATUS
approved