Glenn G. Chappell - cp's OEIS frontend

User: Glenn G. Chappell

Glenn G. Chappell has authored 2 sequences.

A180636 Positive integers that are divisible by neither 8k-1 nor 8k+1, for all k > 0.

1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 13, 16, 19, 20, 22, 24, 26, 29, 32, 37, 38, 40, 43, 44, 48, 52, 53, 58, 59, 61, 64, 67, 74, 76, 80, 83, 86, 88, 96, 101, 104, 106, 107, 109, 116, 118, 122, 128, 131, 134, 139, 148, 149, 152, 157, 160, 163, 166, 172, 173, 176, 179, 181, 192

Offset: 1

Glenn G. Chappell, Sep 13 2010

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A047424. - Robert G. Wilson v, Oct 06 2010

fQ[n_] := Union[ MemberQ[{1, 7}, # ] & /@ Union@ Mod[ Rest@ Divisors@ n, 8]] == {False}; fQ[1] = True; Select[ Range@ 200, fQ] (* Robert G. Wilson v, Oct 06 2010 *)

# Works in Python 2 or 3
import itertools
for n in itertools.count(1):
    for k in range(1, 2+n//8):
        if n%(8*k-1)==0 or n%(8*k+1)==0:
            break
    else:
        print(n)

More terms from Robert G. Wilson v, Oct 06 2010

A033472 Number of n-vertex labeled graphs that are gracefully labeled trees.

Original entry on oeis.org

1, 1, 2, 4, 12, 40, 164, 752, 4020, 23576, 155632, 1112032, 8733628, 73547332, 670789524, 6502948232, 67540932632, 740949762580, 8634364751264, 105722215202120, 1366258578159064, 18468456090865364, 262118487952306820

Offset: 1

Glenn G. Chappell

nonn

A graph with n edges is graceful if its vertices can be labeled with distinct integers in the range 0,1,...,n in such a way that when the edges are labeled  with the absolute differences between the labels of their end-vertices, the n edges have the distinct labels 1,2,...,n.
The PARI/GP program below uses the Matrix-Tree Theorem and sums over {1,-1} vectors to isolate the multilinear term. It takes time 2^n * n^O(1) to compute graceful_tree_count(n). - Noam D. Elkies, Nov 18 2002
Noam D. Elkies and I have independently verified the existing terms and computed more, up to a(31). - Don Knuth, Feb 09 2021

			For n=3 we have 1-3-2 and 2-1-3, so a(3)=2.

A. Kotzig, Recent results and open problems in graceful graphs, Congressus Numerantium, 44 (1984), 197-219 (esp. 200, 204).

Don Knuth and Noam Elkies, Table of n, a(n) for n = 1..31
David Anick, Counting graceful labelings of trees: a theoretical and empirical study, Discrete Applied Mathematics 198 (2016), 65-81.
Index entries for sequences related to trees

Cf. A006967, A337274.

{ treedet(v, n) = n=length(v); matdet(matrix(n,n,i,j, if(i-j,-v[abs(i-j)], sum(m=1,n+1,if(i-m,v[abs(i-m)],0))))) }
{ graceful_tree_count(n, s,v,v1,v0)= if(n==1,1, s=0; v1=vector(n-1,m,1); v0=vector(n-1,m,if(m==1,1,0)); for(m=2^(n-2),2^(n-1)-1, v= binary(m) - v0; s = s + (-1)^(v*v1~) * treedet(v1-2*v) ); s/2^(n-2) ) } \\ Noam D. Elkies, Nov 18 2002
for(n=1,18,print1(graceful_tree_count(n),", ")) \\ Example of function call

a(n) = 2 * A337274(n) for n >= 3. - Hugo Pfoertner, Oct 05 2020

cp's OEIS Frontend

User: Glenn G. Chappell

A180636 Positive integers that are divisible by neither 8k-1 nor 8k+1, for all k > 0.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A033472 Number of n-vertex labeled graphs that are gracefully labeled trees.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

PARI

Formula