|
%I #44 Feb 17 2024 08:11:19
%S 1,2,4,7,13,23,40,65,106,169,278,443,705,1117,1760,2692,4151,6314,
%T 9526,14127,20944,30848,45589,66495,96847,140840,204380,293822,425859,
%U 613446,880288,1258349,1794256,2545965,3623774,5123746,7207773,10159163,14273328,19925242,27893419
%N Number of nonaveraging subsets on {1,2,...,n}.
%H Fausto A. C. Cariboni, <a href="/A051013/b051013.txt">Table of n, a(n) for n = 0..80</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/NonaveragingSequence.html">Nonaveraging Sequence</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Salem-Spencer_set">Salem-Spencer set</a>
%H <a href="/index/No#non_averaging">Index entries related to non-averaging sequences</a>
%F a(n) = 2^n - A018788(n). - _David Nacin_, Mar 03 2012
%e The only subset of s = {1,2,3} that contains a 3-term arithmetic progression is s itself, so a(3) = 7.
%t 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 *)
%o (Python)
%o # Prints out all such sets
%o def nonaveragingsets(n):
%o avoid=list()
%o for skip in range(1,(n+1)//2):
%o for start in range (1,n+1-2*skip):
%o avoid.append(set({start,start+skip,start+2*skip}))
%o s=list()
%o for i in range(3):
%o for smallset in comb(range(1,n+1),i):
%o s.append(smallset)
%o for i in range(3,n+1):
%o for temptuple in comb(range(1,n+1),i):
%o tempset=set(temptuple)
%o status=True
%o for avoidset in avoid:
%o if avoidset <= tempset:
%o status=False
%o break
%o if status:
%o s.append(tempset)
%o return s
%o # Counts all such sets
%o def a(n):
%o return len(nonaveragingsets(n)) # _David Nacin_, Mar 03 2012
%Y Cf. A018788.
%Y Row sums of A334187.
%Y First differences give A334893.
%K nonn
%O 0,2
%A _Eric W. Weisstein_
%E More terms from _John W. Layman_, Nov 27 2001
%E a(29)-a(37) from _Donovan Johnson_, Aug 15 2010
%E a(38)-a(40) from _Alois P. Heinz_, Oct 27 2011
|
