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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).

%I A371663 #23 Oct 06 2024 12:44:16
%S A371663 3,4,5,6,8,9,10,12,15,16,18,20,24,30,36,40,45,48,60,72,80,90,120,144,
%T A371663 180,240,360
%N 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).
%C A371663 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.
%C A371663 Subsequence of A018609 (Divisors of 720).
%H A371663 Wikipedia, <a href="https://en.wikipedia.org/wiki/Arithmetic_progression">Arithmetic progression</a>.
%e A371663 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.
%e A371663 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.
%e A371663 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.
%p A371663 A371663:=proc(k)
%p A371663 	if (k-2)*180/k=floor((k-2)*180/k) then
%p A371663 		return k;
%p A371663 	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
%p A371663 		return k;
%p A371663 	fi;
%p A371663 end proc;
%p A371663 seq(A371663(k),k=3..360);
%Y A371663 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).
%K A371663 fini,full,nonn
%O A371663 1,1
%A A371663 _Felix Huber_, Apr 04 2024