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).
Original entry on oeis.org
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
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.
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).
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.
Original entry on oeis.org
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
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.
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).
-
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);
Comments