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

3, 6, 10, 17, 24, 27, 30, 34, 41, 48, 51, 54, 58, 65, 72, 75, 78, 82, 89, 96, 99, 102, 106, 113, 120, 123, 126, 130, 137, 144, 147, 150, 154, 161, 168, 171, 174, 178, 185, 192, 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

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

Table of n, a(n) for n=1..68.

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

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

Sequence in context: A291986 A360889 A094272 * A308699 A286304 A005045

Adjacent sequences: A236323 A236324 A236325 * A236327 A236328 A236329

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