A235088 a(n)*Pi is the total length of irregular spiral (center points: 1, 2, 3, 4) after n rotations.

3, 6, 17, 28, 47, 66, 93, 120, 155, 190, 233, 276, 327, 378, 437, 496, 563, 630, 705, 780, 863, 946, 1037, 1128, 1227, 1326, 1433, 1540, 1655, 1770, 1893, 2016, 2147, 2278, 2417, 2556, 2703, 2850, 3005, 3160, 3323, 3486, 3657, 3828, 4007, 4186, 4373, 4560, 4755, 4950, 5153, 5356, 5567, 5778, 5997, 6216, 6443, 6670, 6905, 7140

Offset

1,1

Comments

Let points 1, 2, 3 & 4 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 and maintain continuity of circumferences. Continue using this procedure at point 3, 4, 1, ... and so on. The form is expanded spiral. See illustration in links.

Links

Table of n, a(n) for n=1..60.

Kival Ngaokrajang, Illustration of initial terms

Formula

a(n) = 2*floor((n-1)^2/4) + 3*ceiling(n^2/2) (conjectured). - Ralf Stephan, Jan 13 2014

Conjecture: a(n) = 1-(-1)^n-n+2*n^2. a(n) = 2*a(n-1)-2*a(n-3)+a(n-4). G.f.: -x*(5*x^2+3)/((x-1)^3*(x+1)). - Colin Barker, Jan 16 2014

Prog

(Small Basic)
a[1]=3
d1=3
For n = 1 To 100
  If Math.Remainder(n+3, 2)=1 then
    d1=d1+8
  EndIf
  a[n+1]=a[n]+d1
  TextWindow.Write(a[n]+", ")
EndFor

Crossrefs

Cf. A014105*Pi (total spiral length, 2 inline center points). A234902*Pi, A234903*Pi, A234904*Pi (total spiral length, 3 inline center points).

Sequence in context: A217084 A024823 A024315 * A369707 A327068 A307604

Adjacent sequences: A235085 A235086 A235087 * A235089 A235090 A235091

Keyword

nonn

Author

Kival Ngaokrajang, Jan 03 2014

Status

approved