A236327 a(n)*Pi is the total length of irregular spiral (center points: 1, 2, 3, 4, 5; pattern 2) after n rotations.

2, 8, 19, 30, 32, 38, 49, 60, 62, 68, 79, 90, 92, 98, 109, 120, 122, 128, 139, 150, 152, 158, 169, 180, 182, 188, 199, 210, 212, 218, 229, 240, 242, 248, 259, 270, 272, 278, 289, 300, 302, 308, 319, 330, 332, 338, 349, 360, 362, 368, 379, 390, 392, 398, 409, 420, 422, 428, 439, 450, 452, 458, 469, 480, 482, 488

Offset

1,1

Comments

Let points 1, 2, 3, 4 & 5 be placed on a straight line at intervals of 1 unit. At point 1 make a half unit circle then at point 2 make another half circle; by selecting radius point on the right hand side of point 1 (pattern 2); at point 3 make another half circle and maintain continuity of circumferences. Continue using this procedure at point 4, 5, 1, ... and so on.

Conjecture: All forms of 120 permutations 5 center points are non-expanded loops.

Links

Table of n, a(n) for n=1..66.

Kival Ngaokrajang, Illustration of irregular spiral (center points: 1, 2, 3, 4, 5)Pattern 2.

Formula

Conjecture from Colin Barker, Jul 12 2014: (Start)

a(n) = a(n-1)+a(n-4)-a(n-5).

G.f.: x*(11*x^3+11*x^2+6*x+2) / ((x-1)^2*(x+1)*(x^2+1)). (End)

Prog

(Small Basic)
n =5      'center points number 1<=n<=9
pt=1      'pattern1:  pt=-1; pattern2: pt=1
i=12345   'center points order
rota=100  'rotations
sum=0
rc=1
r[1]=1
For i1 = 1 To n
  d1=i/Math.Power(10, 1)
  i=math.Floor(d1)
  d[i1]=(d1-i)*Math.Power(10, 1)
EndFor
For j1=1 To n
  For j2=1 To n
    If d[j1]=j2 Then
      dd[j2]=j1
    endif
  EndFor
EndFor
For j3=1 To n
  If j3=n Then
    dxy[j3]=dd[j3]-dd[1]
  Else
    dxy[j3]=dd[j3]-dd[j3+1]
  EndIf
EndFor
For k1=1 To rota*n
  cc=Math.Floor((k1-1)/n)
  p[k1]=r[k1]+pt*dxy[k1-cc*n]*Math.Power(-1, Math.Remainder(k1, 2))
  r[k1+1]=p[k1]
  sum=sum+math.Abs(r[k1])
  If math.Abs(r[k1])>0 Then
    rc=rc+1
  EndIf
  If rc=3 Then
    TextWindow.Write(sum+", ")
    rc=1
  EndIf
EndFor

Crossrefs

Cf. A014105 (2 center points); A234902, A234903, A234904 (3 center points); A235088, A235089 (4 center points).

Sequence in context: A093012 A049525 A286263 * A109071 A196134 A256321

Adjacent sequences: A236324 A236325 A236326 * A236328 A236329 A236330

Keyword

nonn

Author

Kival Ngaokrajang, Jan 22 2014

Status

approved