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A234902 a(n)*Pi is the total length of irregular spiral (center points: 1, 2, 3) after n rotations.

%I A234902 #31 May 21 2021 18:32:55
%S A234902 2,9,13,17,24,26,33,37,41,48,50,57,61,65,72,74,81,85,89,96,98,105,109,
%T A234902 113,120,122,129,133,137,144,146,153,157,161,168,170,177,181,185,192,
%U A234902 194,201,205,209,216,218,225,229,233,240,242,249,253,257
%N A234902 a(n)*Pi is the total length of irregular spiral (center points: 1, 2, 3) after n rotations.
%C A234902 Let points 1, 2 & 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 the spiral is a non-expanded loop.
%C A234902 The sequence will be A047622 if the second radius = 2; if the second radius = 0, the sequence is a(n).
%C A234902 See illustration in links.
%H A234902 Harvey P. Dale, <a href="/A234902/b234902.txt">Table of n, a(n) for n = 1..1000</a>
%H A234902 Kival Ngaokrajang, <a href="/A234902/a234902_1.pdf">Illustration of initial terms</a>
%H A234902 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,1,-1).
%F A234902 G.f.: x*(7*x^4 + 4*x^3 + 4*x^2 + 7*x + 2)/((1-x)*(1-x^5)). - _Ralf Stephan_, Jan 20 2014
%t A234902 LinearRecurrence[{1,0,0,0,1,-1},{2,9,13,17,24,26},60] (* _Harvey P. Dale_, May 21 2021 *)
%o A234902 (Small Basic)
%o A234902 a[1]=2
%o A234902 For n = 1 To 100
%o A234902   d1=2
%o A234902   m5=math.Remainder(n+1,5)
%o A234902   If m5=0 Or m5=2 Then
%o A234902     d1=7
%o A234902   EndIf
%o A234902   If m5=3 Or m5=4 Then
%o A234902     d1=4
%o A234902   EndIf
%o A234902   a[n+1]=a[n]+d1
%o A234902   TextWindow.Write(a[n]+", ")
%o A234902 EndFor
%Y A234902 Cf. A014105*Pi (total spiral length, 2 inline center points).
%K A234902 nonn,easy
%O A234902 1,1
%A A234902 _Kival Ngaokrajang_, Jan 01 2014