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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
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
Sequence in context: A240991 A304050 A351220 * A030568 A017593 A335908
KEYWORD
nonn
AUTHOR
Kival Ngaokrajang, May 13 2014
STATUS
approved

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Last modified May 30 03:30 EDT 2024. Contains 372957 sequences. (Running on oeis4.)