A234904 a(n)*Pi is the total length of irregular spiral (center points: 2, 1, 3) after n rotations.

3, 12, 18, 21, 30, 36, 39, 48, 54, 57, 66, 72, 75, 84, 90, 93, 102, 108, 111, 120, 126, 129, 138, 144, 147, 156, 162, 165, 174, 180, 183, 192, 198, 201, 210, 216, 219, 228, 234, 237, 246, 252, 255, 264, 270, 273, 282, 288, 291, 300, 306, 309, 318, 324, 327, 336, 342, 345, 354, 360, 363, 372, 378, 381, 390, 396, 399, 408

Offset

1,1

Comments

Let points 2, 1 & 3 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 points 3, 1, 2, and so on. The form of spiral is non-expanded loop. See illustration in links.

Links

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

Sela Fried, Lengths of curves formed from successive semicircles, 2026.

Kival Ngaokrajang, Illustration of initial terms

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1)

Formula

a(n) = 3*A047234(n+1).

From Colin Barker, Jul 12 2014: (Start)

a(n) = a(n-1) + a(n-3) - a(n-4).

G.f.: 3*x*(x+1)*(2*x+1) / ((x-1)^2*(x^2+x+1)). (End)

Interlaced polynomials: a(3n) = 18*n; a(3n+1) = 18*n+3; a(3n+2) = 18*n + 12 for n > 0. - Avi Friedlich, May 16 2015

a(n) = 6*n - 3*floor((n+2)/3) + 3*floor((n+1)/3). - Sela Fried, May 08 2026

Mathematica

RecurrenceTable[{a[n] == a[n - 1] + a[n - 3] - a[n - 4], a[1] == 3,
a[2] == 12, a[3] == 18, a[4] == 21}, a, {n, 1, 68}] (* Michael De Vlieger, May 09 2015 *)
LinearRecurrence[{1, 0, 1, -1}, {3, 12, 18, 21}, 70] (* Vincenzo Librandi, May 10 2015 *)

Prog

(Small Basic)
a[1]=3
For n = 1 To 100
  d1=3
  m3 = math.Remainder(n+1, 3)
  If m3 = 0 Then
    d1 = 6
  EndIf
  If m3 = 2 Then
    d1 = 9
  EndIf
  a[n+1]=a[n]+d1
  TextWindow.Write(a[n]+", ")
EndFor
(PARI) Vec(3*x*(x+1)*(2*x+1)/((x-1)^2*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Jul 12 2014
(Magma) I:=[3, 12, 18, 21]; [n le 4 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..70]]; // Vincenzo Librandi, May 10 2015

Crossrefs

Cf. A014105*Pi (total spiral length, 2 inline center points).

Sequence in context: A342776 A104641 A032703 * A284806 A379794 A125821

Adjacent sequences: A234901 A234902 A234903 * A234905 A234906 A234907

Keyword

nonn,easy

Author

Kival Ngaokrajang, Jan 01 2014

Status

approved