A051013 Number of nonaveraging subsets on {1,2,...,n}.

1, 2, 4, 7, 13, 23, 40, 65, 106, 169, 278, 443, 705, 1117, 1760, 2692, 4151, 6314, 9526, 14127, 20944, 30848, 45589, 66495, 96847, 140840, 204380, 293822, 425859, 613446, 880288, 1258349, 1794256, 2545965, 3623774, 5123746, 7207773, 10159163, 14273328, 19925242, 27893419

Offset: 0

Author

Eric W. Weisstein

Keywords

• nonn

Examples

			The only subset of s = {1,2,3} that contains a 3-term arithmetic progression is s itself, so a(3) = 7.
		

Links

• Fausto A. C. Cariboni, Table of n, a(n) for n = 0..80
• Eric Weisstein's World of Mathematics, Nonaveraging Sequence
• Wikipedia, Salem-Spencer set
• Index entries related to non-averaging sequences

Crossrefs

Cf. A018788.

Row sums of A334187.

First differences give A334893.

Programs

Mathematica

a[n_] := a[n] = 2^n - Count[Subsets[Range[n], {3, n}], {_, a_, _, b_, _, c_, _} /; b-a == c-b]; Table[Print[n, " ", a[n]]; a[n], {n, 0, 32}] (* Jean-François Alcover, May 30 2019 *)

Python

# Prints out all such sets
def nonaveragingsets(n):
    avoid=list()
    for skip in range(1,(n+1)//2):
        for start in range (1,n+1-2*skip):
            avoid.append(set({start,start+skip,start+2*skip}))
    s=list()
    for i in range(3):
        for smallset in comb(range(1,n+1),i):
            s.append(smallset)
    for i in range(3,n+1):
        for temptuple in comb(range(1,n+1),i):
            tempset=set(temptuple)
            status=True
            for avoidset in avoid:
                if avoidset <= tempset:
                    status=False
                    break
            if status:
                s.append(tempset)
    return s
# Counts all such sets
def a(n):
    return len(nonaveragingsets(n)) # David Nacin, Mar 03 2012

Formula

a(n) = 2^n - A018788(n). - David Nacin, Mar 03 2012

Extensions

More terms from John W. Layman, Nov 27 2001

a(29)-a(37) from Donovan Johnson, Aug 15 2010

a(38)-a(40) from Alois P. Heinz, Oct 27 2011