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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. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A318290 A291986 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

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Last modified October 16 07:30 EDT 2019. Contains 328051 sequences. (Running on oeis4.)