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
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
KEYWORD
nonn
AUTHOR
Kival Ngaokrajang, Jan 22 2014
STATUS
approved