A371663 - OEIS

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

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OFFSET

1,1

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).

LINKS

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

Wikipedia, Arithmetic progression.

EXAMPLE

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.

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);

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).

Sequence in context: A100966 A358973 A063977 * A290136 A359168 A332739

Adjacent sequences: A371660 A371661 A371662 * A371664 A371665 A371666

KEYWORD

fini,full,nonn

AUTHOR

Felix Huber, Apr 04 2024

STATUS

approved