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

%I A236327 #13 Jul 12 2014 16:46:03
%S A236327 2,8,19,30,32,38,49,60,62,68,79,90,92,98,109,120,122,128,139,150,152,
%T A236327 158,169,180,182,188,199,210,212,218,229,240,242,248,259,270,272,278,
%U A236327 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
%N A236327 a(n)*Pi is the total length of irregular spiral (center points: 1, 2, 3, 4, 5; pattern 2) after n rotations.
%C A236327 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.
%C A236327 Conjecture: All forms of 120 permutations 5 center points are non-expanded loops.
%H A236327 Kival Ngaokrajang, <a href="/A236327/a236327.pdf">Illustration of irregular spiral (center points: 1, 2, 3, 4, 5)</a>Pattern 2.
%F A236327 Conjecture from _Colin Barker_, Jul 12 2014: (Start)
%F A236327 a(n) = a(n-1)+a(n-4)-a(n-5).
%F A236327 G.f.: x*(11*x^3+11*x^2+6*x+2) / ((x-1)^2*(x+1)*(x^2+1)). (End)
%o A236327 (Small Basic)
%o A236327 n =5      'center points number 1<=n<=9
%o A236327 pt=1      'pattern1:  pt=-1; pattern2: pt=1
%o A236327 i=12345   'center points order
%o A236327 rota=100  'rotations
%o A236327 sum=0
%o A236327 rc=1
%o A236327 r[1]=1
%o A236327 For i1 = 1 To n
%o A236327   d1=i/Math.Power(10,1)
%o A236327   i=math.Floor(d1)
%o A236327   d[i1]=(d1-i)*Math.Power(10,1)
%o A236327 EndFor
%o A236327 For j1=1 To n
%o A236327   For j2=1 To n
%o A236327     If d[j1]=j2 Then
%o A236327       dd[j2]=j1
%o A236327     endif
%o A236327   EndFor
%o A236327 EndFor
%o A236327 For j3=1 To n
%o A236327   If j3=n Then
%o A236327     dxy[j3]=dd[j3]-dd[1]
%o A236327   Else
%o A236327     dxy[j3]=dd[j3]-dd[j3+1]
%o A236327   EndIf
%o A236327 EndFor
%o A236327 For k1=1 To rota*n
%o A236327   cc=Math.Floor((k1-1)/n)
%o A236327   p[k1]=r[k1]+pt*dxy[k1-cc*n]*Math.Power(-1,Math.Remainder(k1,2))
%o A236327   r[k1+1]=p[k1]
%o A236327   sum=sum+math.Abs(r[k1])
%o A236327   If math.Abs(r[k1])>0 Then
%o A236327     rc=rc+1
%o A236327   EndIf
%o A236327   If rc=3 Then
%o A236327     TextWindow.Write(sum+", ")
%o A236327     rc=1
%o A236327   EndIf
%o A236327 EndFor
%Y A236327 Cf. A014105 (2 center points); A234902, A234903, A234904 (3 center points); A235088, A235089 (4 center points).
%K A236327 nonn
%O A236327 1,1
%A A236327 _Kival Ngaokrajang_, Jan 22 2014