A341584 - OEIS

%I #24 Feb 21 2021 14:08:33

%S 0,1,2,2,2,2,4,3,4,4,4,4,4,4,6,4,6,5,8,6,8,6,8,6,8,7,8,8,8,8,8,8,9,8,

%T 10,9,10,10,12,10,11,9,12,9,12,10,12,10,13,10,14,11,14,11,16,11,16,12,

%U 16,12,16,13,16,13,16,14,16,13,16,14,18,14,16,14,20,14,16,14,20,15,18

%N Size of the largest subset of the numbers [1..n] which does not contain a 3-term arithmetic progression modulo n.

%C This is similar to A003002, but the arithmetic progression is modulo n here.

%C For n >= 3, a(n) can be viewed as the maximum number of vertices that can be chosen from a regular polygon with n sides so that no three of them form an isosceles or equilateral triangle.

%H Math StackExchange, <a href="https://math.stackexchange.com/questions/4012158/number-of-points-chosen-form-a-polygon-to-have-no-isosceles-and-equilateral-tria">A question about this sequence</a>.

%e n, a(n), example of an optimal subset:

%e 0, 0, {}

%e 1, 1, {1}

%e 2, 2, {1,2}

%e 3, 2, {1,2}

%e 4, 2, {1,2}

%e 5, 2, {1,2}

%e 6, 4, {1,2,4,5}

%e 7, 3, {1,2,4}

%e 8, 4, {1,2,4,5}

%e 9, 4, {1,2,4,5}

%e 10, 4, {1,2,4,5}

%e 11, 4, {1,2,4,5}

%e 12, 4, {1,2,4,5}

%e 13, 4, {1,2,4,5}

%e 14, 6, {1,2,4,8,9,11}

%e 15, 4, {1,2,4,5}

%e 16, 6, {1,3,4,8,9,11}

%e 17, 5, {1,2,6,7,9}

%e 18, 8, {1,2,4,5,10,11,13,14}

%e 19, 6, {1,3,4,8,9,11}

%e 20, 8, {1,2,4,5,11,12,14,15}

%o (C) /* For n >= 3: */

%o int a(int n)

%o {

%o int upp, maxb, s, i, l, h, maxs;

%o uint64_t b, bs, m, mv[31], mn;

%o for (l = 1; l <= 31; l++) { mv[l - 1] = 1 << 2*l; mv[l - 1] |= (1 << l); mv[l - 1] |= 1; }

%o maxb = 1 << n; mn = maxb - 1; h = (n - 1) / 2; maxs = 2*n/3; upp = 0;

%o for (b = 0; b < maxb; b++) {

%o   for (i = 0, s = 0, m = 1; i < n; i++, m <<= 1) { if (b & m) s++; }

%o   if (s <= maxs) {

%o    for (l = 1; l <= h; l++) {

%o     m = mv[l - 1];

%o     for (i = 0; i < n; i++) { if ((b & m) == m) { l = 1000; break; } m = ((m << 1) & mn) | (m >> (n - 1)); }

%o    }

%o    if (l < 1000 && s > upp) upp = s;

%o   }

%o }

%o return upp;

%o }

%Y Cf. A003002.

%K nonn,more

%O 0,3

%A _Fabio Visonà_, Feb 15 2021

%E a(31)-a(80) from _Bert Dobbelaere_, Feb 20 2021