OFFSET
1,1
COMMENTS
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.
Also floor(a(n)/3) = A047607(n+1).
Conjecture: All forms of 120 permutations of 5 center points are non-expanded loops.
LINKS
Kival Ngaokrajang, Illustration of irregular spirals (center points: 1, 2, 3, 4, 5) Pattern 1.
FORMULA
Conjecture from Colin Barker, Jul 12 2014: (Start)
a(n) = a(n-1)+a(n-5)-a(n-6).
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)
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
EXTENSIONS
Description of procedure for constructing curve (under Comments) edited by Jon E. Schoenfield, Feb 12 2014
STATUS
approved