A383968 - OEIS

%I #23 May 29 2025 00:12:01

%S 1,1,2,3,5,9,17,30,58,107,205,392,768,1466,2883,5597,11038,21572,

%T 42675,83711,166371,327893,651199,1288480,2564032,5082878,10127472,

%U 20115845,40104636,79781149,159174500,316962113,632716744,1261189166,2518287361,5023170116,10034132101,20025033970

%N Number of distinct subsets S of [1..n] such that for all 1 <= k <= n, there exists two elements x,y in S (not necessarily distinct) such that x+y = 2k.

%C Every subset S of [1..n] must have 1 and n to get 2 and 2*n. For odd n we therefore have 1+n which we need as well. If S is no such subset then no subset S' of S must be tested. - _David A. Corneth_, May 22 2025

%H David A. Corneth, <a href="/A383968/a383968.gp.txt">PARI program</a>

%H Sean A. Irvine, <a href="https://github.com/archmageirvine/joeis/blob/master/src/irvine/oeis/a383/A383968.java">Java program</a> (github)

%e For n = 5, there are 5 sets S that satisfy the said conditions: {1, 2, 3, 4, 5}, {1, 2, 3, 5}, {1, 2, 4, 5}, {1, 3, 4, 5} and {1, 3, 5}.

%o (Python)

%o def a(n):

%o     if n == 1: return 1

%o     c,t = 0, set(k << 1 for k in range(1, n+1))

%o     for i in range(1 << (n-2), 1 << n):

%o         s = [j+1 for j in range(n) if (i >> j) & 1]

%o         if s[0] == 1 and s[-1] == n:

%o             ss = set(x + y for x in s for y in s if x & 1 == y & 1)

%o             if t.issubset(ss): c += 1

%o     return c # _Darío Clavijo_, May 23 2025

%K nonn

%O 1,3

%A _SiYang Hu_, May 16 2025

%E a(23)-a(38) from _Sean A. Irvine_, May 21 2025