A242394 Number of equilateral triangles (sides length = 1) that intersect the circumference of a circle of radius n centered at (0,0).

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

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: A304050 A376028 A351220 * A030568 A017593 A335908

Adjacent sequences: A242391 A242392 A242393 * A242395 A242396 A242397

Keyword

nonn

Author

Kival Ngaokrajang, May 13 2014

Status

approved