cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-2 of 2 results.

A333728 Maximum number of graceful labelings for a simple graph on n nodes.

Original entry on oeis.org

1, 2, 12, 48, 168, 1152, 9600, 97920
Offset: 1

Views

Author

Eric W. Weisstein, Apr 03 2020

Keywords

Comments

a(9) >= 1491840 (K_1,1,7).
Table of graphs achieving these maxima:
1: K_1
2: K_2 = P_2
3: K_3 = C_3 = K_1,1,1
4: K_4
5: K_1,1,3
6: K_1,1,4
7: K_1,1,5
8: K_1,1,6
a(4) fails to correspond to the diamond graph K_1,1,2 since K_4 has 48 graceful labelings, while K_1,1,2 has only 32.

Crossrefs

Cf. A334307, A339892.

Extensions

a(8) from Eric W. Weisstein, Jul 30 2020

A339891 Number of fundamentally different graceful labelings of the complete tripartite graph K_{1,1,n}.

Original entry on oeis.org

1, 4, 7, 12, 20, 34, 74, 131, 260, 524, 1030, 2054, 4118, 8196, 16389, 32804, 65554, 131074, 262216, 524292, 1048580, 2097304, 4194312, 8388619, 16777478, 33554436, 67108906, 134218244, 268435464, 536870914, 1073742880, 2147483720, 4294967300, 8589936646, 17179869193
Offset: 1

Views

Author

Don Knuth, Dec 21 2020

Keywords

Comments

The difference between "fundamentally different graceful labelings" of a graph and "graceful labelings" of a graph is that the latter is the former multiplied by twice the number of automorphisms. (The extra factor of 2 comes from complementation.)
When n>1, the graph K_{1,1,n} has 2n! automorphisms.

References

  • D. E. Knuth, The Art of Computer Programming, Section 7.2.2.3, in preparation.

Links

Crossrefs

If n>1, A334307(n) = 4*a(n)*n!.

Programs

  • Mathematica
    A339891[n_]:=If[n==1,1,DivisorSum[2n+1,2^((#-1)/2)&]+DivisorSigma[0,n+1]-2^(n-1)-1];Array[A339891,50] (* Paolo Xausa, Dec 04 2023 *)

Formula

a(n) = A339916(n) + A000005(n+1) - 2^(n-1) - 1 - 2*[n=1].
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.