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.
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
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
- Wikipedia, Arithmetic sequence
Crossrefs
Programs
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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);
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