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A236326 a(n)*Pi is the total length of irregular spiral (center points: 1, 2, 3, 4, 5; pattern 1) after n rotations.

%I A236326 #19 Jul 12 2014 12:19:24
%S A236326 3,6,10,17,24,27,30,34,41,48,51,54,58,65,72,75,78,82,89,96,99,102,106,
%T A236326 113,120,123,126,130,137,144,147,150,154,161,168,171,174,178,185,192,
%U A236326 195,198,202,209,216,219,222,226,233,240,243,246,250,257,264,267,270,274,281,288,291,294,298,305,312,315,318,322
%N A236326 a(n)*Pi is the total length of irregular spiral (center points: 1, 2, 3, 4, 5; pattern 1) after n rotations.
%C A236326 Let points 1, 2, 3, 4, and 5 be placed on a horizontal straight line at intervals of 1 unit. (See illustration under Links.) Construct a continuous curve from half circles, alternately lying below and above the line, centered at points 1, 2, 3, 4, 5, 1, ... and so on, with the first half circle (centered at point 1) beginning at point 2.
%C A236326 Also floor(a(n)/3) = A047607(n+1).
%C A236326 Conjecture: All forms of 120 permutations of 5 center points are non-expanded loops.
%H A236326 Kival Ngaokrajang, <a href="/A236326/a236326.pdf">Illustration of irregular spirals (center points: 1, 2, 3, 4, 5)</a> Pattern 1.
%F A236326 Conjecture from _Colin Barker_, Jul 12 2014: (Start)
%F A236326 a(n) = a(n-1)+a(n-5)-a(n-6).
%F A236326 G.f.: x*(7*x^4+7*x^3+4*x^2+3*x+3) / ((x-1)^2*(x^4+x^3+x^2+x+1)). (End)
%o A236326 (Small Basic)
%o A236326 n =5       'center points number 1<=n<=9
%o A236326 pt=-1      'pattern1: pt=-1; pattern2: pt=1
%o A236326 i=12345    'center points order
%o A236326 rota=100   'rotations
%o A236326 sum=0
%o A236326 rc=1
%o A236326 r[1]=1
%o A236326 For i1 = 1 To n
%o A236326   d1=i/Math.Power(10,1)
%o A236326   i=math.Floor(d1)
%o A236326   d[i1]=(d1-i)*Math.Power(10,1)
%o A236326 EndFor
%o A236326 For j1=1 To n
%o A236326   For j2=1 To n
%o A236326     If d[j1]=j2 Then
%o A236326       dd[j2]=j1
%o A236326     endif
%o A236326   EndFor
%o A236326 EndFor
%o A236326 For j3=1 To n
%o A236326   If j3=n Then
%o A236326     dxy[j3]=dd[j3]-dd[1]
%o A236326   Else
%o A236326     dxy[j3]=dd[j3]-dd[j3+1]
%o A236326   EndIf
%o A236326 EndFor
%o A236326 For k1=1 To rota*n
%o A236326   cc=Math.Floor((k1-1)/n)
%o A236326   p[k1]=r[k1]+pt*dxy[k1-cc*n]*Math.Power(-1,Math.Remainder(k1,2))
%o A236326   r[k1+1]=p[k1]
%o A236326   sum=sum+math.Abs(r[k1])
%o A236326   If math.Abs(r[k1])>0 Then
%o A236326     rc=rc+1
%o A236326   EndIf
%o A236326   If rc=3 Then
%o A236326     TextWindow.Write(sum+", ")
%o A236326     rc=1
%o A236326   EndIf
%o A236326 EndFor
%Y A236326 Cf. A014105 (2 center points); A234902, A234903, A234904 (3 center points); A235088, A235089 (4 center points).
%K A236326 nonn
%O A236326 1,1
%A A236326 _Kival Ngaokrajang_, Jan 22 2014
%E A236326 Description of procedure for constructing curve (under Comments) edited by _Jon E. Schoenfield_, Feb 12 2014