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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: A077660 A240991 A304050 * 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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Last modified May 6 18:38 EDT 2021. Contains 343586 sequences. (Running on oeis4.)