cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A100659 Floor of measure (in degrees) of the internal angles of a regular polygon with n sides.

Original entry on oeis.org

60, 90, 108, 120, 128, 135, 140, 144, 147, 150, 152, 154, 156, 157, 158, 160, 161, 162, 162, 163, 164, 165, 165, 166, 166, 167, 167, 168, 168, 168, 169, 169, 169, 170, 170, 170, 170, 171, 171, 171, 171, 171, 172, 172, 172, 172, 172, 172, 172, 173, 173, 173, 173, 173
Offset: 3

Views

Author

Federico Arboleda (federico.arboleda(AT)gmail.com), Jan 02 2005

Keywords

Examples

			a(4) = 90 because all angles of a square measure 90 degrees.

Crossrefs

Cf. A069976.

Programs

Formula

a(n) = floor(180 - 360/n). - Mark Hudson.

Extensions

Edited by Mark Hudson, Jan 05 2005
First term 0 removed by Michel Marcus, Aug 27 2013

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

Views

Author

Felix Huber, Apr 04 2024

Keywords

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).

Examples

			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.

Links

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).

Programs

  • 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);

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

Views

Author

Felix Huber, Apr 04 2024

Keywords

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.

Examples

			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.

Links

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).

Programs

  • 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);

A110546 In degrees, values of interior angles of regular polygons whose angles are integers.

Original entry on oeis.org

60, 90, 108, 120, 135, 140, 144, 150, 156, 160, 162, 165, 168, 170, 171, 172, 174, 175, 176, 177, 178, 179
Offset: 1

Views

Author

Alexandre Wajnberg, Sep 11 2005

Keywords

Comments

The number of sides for which the interior angles in degrees are integers is given by A018412 (except the first two terms).
Integers of the form (k-2)*180/k where k >= 3. - Jason Yuen, Sep 05 2024

Examples

			a(3)=108, i.e., the value (in degrees) of the interior angles of the third polygon whose angles are integers (the pentagon).

Crossrefs

Cf. A069976, A100659, A018412.

Formula

a(n) = (A018412(n+2)-2)*180/A018412(n+2). - Jason Yuen, Sep 05 2024

A110547 Number of sides of regular polygons whose interior angles (in degrees) are not integers.

Original entry on oeis.org

7, 11, 13, 14, 16, 17, 19, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86
Offset: 1

Views

Author

Alexandre Wajnberg, Sep 11 2005

Keywords

Comments

Numbers that are not divisors of 360.

Examples

			a(3)=13 because the third polygon whose interior angle in degrees is not an integer is the 13-gon.

Crossrefs

Cf. A069976, A100659, A018412.

Programs

  • Mathematica
    Select[Range[86],!Divisible[360,#] &] (* Stefano Spezia, Sep 05 2024 *)
Showing 1-5 of 5 results.

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