A371664 a(n) is the number of arithmetic progressions that can be formed from all the interior angles (all integers when measured in degrees) of a regular polygon with A371663(n) sides.

60, 30, 54, 24, 20, 35, 16, 14, 23, 10, 10, 9, 8, 6, 5, 5, 8, 4, 3, 3, 2, 2, 2, 1, 1, 1, 1

Offset

1,1

Comments

Since A371663 is finite, this sequence is also finite.

With all interior angles (integers when measured in degrees) of simple polygons, two geometric progressions (see comments in A000244 and A007283 from Feb 15 2024) and 357 arithmetic progressions are possible.

Links

Table of n, a(n) for n=1..27.

Wikipedia, Arithmetic sequence

Example

Since A371663(17) = 45 and from a 45-gon 8 arithmetic progressions p_i(k) can formed from all its interior angles (all integer, in degrees), a(17) = 8. The 8 sequences are: p_1(k) = 172, p_2(k) = k + 150, p_3(k) = 2k + 128, p_4(k) = 3k + 106, p_5(k) = 4k + 84, p_6(k) = 5k + 62, p_7(k) = 6k + 40, p_8(k) = 7k + 18, for integers k with 0 <= k <= 44.
Since A371663(19) = 60 and from a 60-gon 3 arithmetic progressions p_i(k) can formed from all its interior angles (all integer, in degrees), a(19) = 3. The 3 sequences are: p_1(k) = 174, p_2(k) = 2k + 115, p_3(k) = 4k + 56, for integers k with 0 <= k <= 15.
Since A371663(10) = 16 and from a 16-gon 10 arithmetic progressions p_i(k) can formed from all its interior angles (all integer, in degrees), a(10) = 10. The 10 sequences are: p_1(k) = k + 150, p_2(k) = 3k + 135, p_3(k) = 5k + 120, p_4(k) = 7k + 105, p_5(k) = 9k + 90, p_6(k) = 11k + 75, p_7(k) = 13k + 60, p_8(k) = 15k + 45, p_9(k) = 17k + 30, p_10(k) = 19k + 15 for integers k with 0 <= k <= 15.

Maple

A371664:=proc(n)
	local a, L;
	L:=[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];
	if (L[n]-2)*180/L[n]=floor((L[n]-2)*180/L[n]) then
		if  L[n] mod 2 = 1 then
			a:=ceil(((L[n]-2)*360/L[n])/(L[n]-1))
		else a:=ceil(((L[n]-2)*180/L[n])/(L[n]-1))
		fi;
	elif (L[n]-2)*360/L[n]=floor((L[n]-2)*360/L[n]) and L[n] mod 2 = 0 then
		a:=ceil(((L[n]-2)*360/L[n]-L[n]+1)/(2*(L[n]-1)))
	fi;
	return a;
end proc;
seq(A371664(n), n=1..27);

Crossrefs

Cf. A371663, A018412 (regular polygons, first comment), A018609 (Divisors of 720), A069976 (interior angle of regular polygons), A000244 (geometric progression, comment from Feb 15 2024), A007283 (geometric progression, comment from Feb 15 2024).

Sequence in context: A133000 A033380 A094086 * A206480 A230742 A112025

Adjacent sequences: A371661 A371662 A371663 * A371665 A371666 A371667

Keyword

fini,full,nonn

Author

Felix Huber, Apr 04 2024

Status

approved