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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.

%I A371664 #7 Apr 28 2024 11:17:08
%S A371664 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
%N 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.
%C A371664 Since A371663 is finite, this sequence is also finite.
%C A371664 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.
%H A371664 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_progression">Arithmetic sequence</a>
%e A371664 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.
%e A371664 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.
%e A371664 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.
%p A371664 A371664:=proc(n)
%p A371664 	local a,L;
%p A371664 	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];
%p A371664 	if (L[n]-2)*180/L[n]=floor((L[n]-2)*180/L[n]) then
%p A371664 		if  L[n] mod 2 = 1 then
%p A371664 			a:=ceil(((L[n]-2)*360/L[n])/(L[n]-1))
%p A371664 		else a:=ceil(((L[n]-2)*180/L[n])/(L[n]-1))
%p A371664 		fi;
%p A371664 	elif (L[n]-2)*360/L[n]=floor((L[n]-2)*360/L[n]) and L[n] mod 2 = 0 then
%p A371664 		a:=ceil(((L[n]-2)*360/L[n]-L[n]+1)/(2*(L[n]-1)))
%p A371664 	fi;
%p A371664 	return a;
%p A371664 end proc;
%p A371664 seq(A371664(n),n=1..27);
%Y A371664 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).
%K A371664 fini,full,nonn
%O A371664 1,1
%A A371664 _Felix Huber_, Apr 04 2024