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 A242394 Number of equilateral triangles (sides length = 1) that intersect the circumference of a circle of radius n centered at (0,0). 4
 6, 18, 30, 42, 54, 66, 66, 102, 114, 126, 138, 150, 150, 162, 198, 210, 222, 234, 222, 270, 258, 294, 306, 318, 330, 330, 366, 354, 390, 402, 390, 426, 450, 462, 450, 486, 474, 486, 510, 546, 558, 546, 558, 594, 606, 630, 642, 654, 618, 678, 690, 690, 726, 738, 750, 738, 750 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS For all n, there are at least 6 points where the transit of circumference occurs exactly at the corners. The rare case is when the transit occurs at 2 corners of a triangle, i.e., at n = 1, 13, 181, 35113, ... , (A001570(n)). The pattern repeats itself at every Pi/3 sector along the circumference. The triangle count per half sector by rows can be arranged as an irregular triangle as shown in the illustration. The rows count (A242396) is equal to the case centered at (1/2,0), A242395. LINKS Table of n, a(n) for n=1..57. Kival Ngaokrajang, Illustration of initial terms Kival Ngaokrajang, Illustration for rare cases PROG (Small Basic) For n =1 To 100 r6=n*math.Sin(30*Math.Pi/180)/(Math.Power(3, 0.5)/2) r6a=math.Round(r6) If r6-math.Floor(r6) >0.5 Then last=1 Else last=2 EndIf 'find corner intersecting points----------------------- k=0 ic=0 h=Math.Power(1-0.5*0.5, 0.5) c=math.Floor(n/h) For i = h To c Step h For j = 0.5 To n Step 0.5 r=Math.Power(i*i+j*j, 0.5) If r = n Then k=k+1 EndIf EndFor EndFor if k > 1 then ic=math.Floor(k/3) EndIf '------------------------------------------------------ a=0 b=0 For ii=1 To r6a If ii=1 Then a=a+1 Else If ii = r6a Then a=a+last Else a=a+2 EndIf EndIf b=a EndFor if n =1 then aa = 1 Else aa =1*(a-2*ic)*2+1 endif TextWindow.Write(6*aa+", ") EndFor CROSSREFS Cf. A001570, A242118. Sequence in context: A240991 A304050 A351220 * A030568 A017593 A335908 Adjacent sequences: A242391 A242392 A242393 * A242395 A242396 A242397 KEYWORD nonn AUTHOR Kival Ngaokrajang, May 13 2014 STATUS approved

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