A371663 a(n) is the number of sides of simple polygons (sorted in ascending order) for which one or more arithmetic sequences can be formed from all their interior angles (all integer, in degrees).

3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360

Offset: 1

Author

Felix Huber, Apr 04 2024

Keywords

• fini
• full
• nonn

Comments

Since for n >= 357 every arithmetic sequence with d != 0 would have a smallest term less than 1 and for n > 360 no regular polygon with integer angles exists (see also A018412), this sequence is finite and contains 27 terms.

Subsequence of A018609 (Divisors of 720).

Examples

			Since the sum of the interior angles of a triangle is 180 degrees and an interior angle is 60 degrees on average, arithmetic sequences 60 - d, 60, 60 + d are possible, for integers d with 0 <= d <= 59. Therefore 3 is a term.
Since the sum of the interior angles of a quadrilateral is 360 degrees and an interior angle is 90 degrees on average, arithmetic sequences 90 - 3d/2, 90 - d/2, 90 + d/2, 90 + 3d/2 are possible, for even d with 0 <= d <= 58. Therefore 4 is a term.
Since the sum of the interior angles of a 16-gon is 2520 degrees and an interior angle is 157.5 degrees on average, arithmetic sequences 157.5 - 15d/2, 157.5 - 13d/2, 157.5 - 11d/2, 157.5 - 9d/2, 157.5 - 7d/2, 157.5 - 5d/2, 157.5 - 3d/2, 157.5 - d/2, 157.5 + d/2, 157.5 + 3d/2, 157.5 + 5d/2, 157.5 + 7d/2, 157.5 + 9d/2, 157.5 + 11d/2, 157.5 + 13d/2, 157.5 + 15d/2 are possible, for odd d with 1 <= d <= 19. Therefore 16 is a term.
		

Links

• Wikipedia, Arithmetic progression.

Crossrefs

Cf. A371664, A018412 (regular polygons, first comment), A018609 (divisors of 720), A069976 (interior angle of regular polygons), A000244 (geometric sequence, comment from Feb 15 2024), A007283 (geometric sequence, comment from Feb 15 2024).

Programs

Maple

A371663:=proc(k)
	if (k-2)*180/k=floor((k-2)*180/k) then
		return k;
	elif (k-2)*360/k=floor((k-2)*360/k) and ceil(((k-2)*360/k-k+1)/(2*(k-1)))>0 and k mod 2 = 0 then
		return k;
	fi;
end proc;
seq(A371663(k),k=3..360);