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A341584 Size of the largest subset of the numbers [1..n] which does not contain a 3-term arithmetic progression modulo n. 0
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, 10, 9, 10, 10, 12, 10, 11, 9, 12, 9, 12, 10, 12, 10, 13, 10, 14, 11, 14, 11, 16, 11, 16, 12, 16, 12, 16, 13, 16, 13, 16, 14, 16, 13, 16, 14, 18, 14, 16, 14, 20, 14, 16, 14, 20, 15, 18 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

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.

LINKS

Table of n, a(n) for n=0..80.

Math StackExchange, A question about this sequence.

EXAMPLE

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

0, 0, {}

1, 1, {1}

2, 2, {1,2}

3, 2, {1,2}

4, 2, {1,2}

5, 2, {1,2}

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

7, 3, {1,2,4}

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

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

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

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

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

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

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

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

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

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

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

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

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

PROG

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

int a(int n)

{

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

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

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

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

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

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

  if (s <= maxs) {

   for (l = 1; l <= h; l++) {

    m = mv[l - 1];

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

   }

   if (l < 1000 && s > upp) upp = s;

  }

}

return upp;

}

CROSSREFS

Cf. A003002.

Sequence in context: A098983 A097576 A029250 * A110884 A303637 A155904

Adjacent sequences:  A341581 A341582 A341583 * A341585 A341586 A341587

KEYWORD

nonn,more

AUTHOR

Fabio VisonĂ , Feb 15 2021

EXTENSIONS

a(31)-a(80) from Bert Dobbelaere, Feb 20 2021

STATUS

approved

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Last modified May 5 23:41 EDT 2021. Contains 343579 sequences. (Running on oeis4.)