%I #17 Jul 12 2014 16:46:10
%S 2,5,8,10,13,16,18,21,24,26,29,32,34,37,40,42,45,48,50,53,56,58,61,64,
%T 66,69,72,74,77,80,82,85,88,90,93,96,98,101,104,106,109,112,114,117,
%U 120,122,125,128,130,133,136,138,141,144,146,149,152,154,157,160,162,165,168,170,173,176,178,181,184,186,189
%N a(n)*Pi is the total length of irregular spiral (center points: 2, 3, 1; pattern 1) after n rotations.
%C Let points 2, 3, & 1 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; by selecting radius point on the left hand side of point 1 (pattern 1); at point 3 make another half circle and maintain continuity of circumferences. Continue using this procedure at point 1, 2, 3, ... and so on.
%C Conjecture: All forms of 3 center points are non-expanded loops.
%C There are other sets of center points that give the same sequence, e.g.: [2,3,1,4]; [3,2,4,1]; [3,2,4,1,5]; [2,3,1,4,5,7,6]; [2,3,1,7,4,6,5]; [3,4,2,5,1,6,7]; [4,3,5,6,2,7,1]; [4,5,3,2,1,6,7]; [5,4,6,3,2,7,1].
%C Also, there are some similar patterns that give difference sequences, e.g.:
%C A047622: [1,2,7,3,4,6,5]; [1,2,7,6,3,5,4]...
%C A047399: [1,2,7,3,6,4,5]; [1,2,7,6,5,3,4]...
%C A047395: [2,3,1,4 7,5,6]; [2,3,1,7,6,4,5]...
%C A047464: [4,5,3,6,2,7,1]; [1,8,2,7,3,6,4,5];
%C [9,1,8,2,7,3,6,4,5].
%C See illustration in links.
%C Appears to be basically a duplicate of A047618. - _R. J. Mathar_, Feb 03 2014
%H Kival Ngaokrajang, <a href="/A236535/a236535.pdf">Illustration of irregular spirals (Center points 3..9)</a>
%F Conjecture from _Colin Barker_, Jul 12 2014: (Start)
%F a(n) = a(n-1)+a(n-3)-a(n-4).
%F G.f.: x*(3*x^2+3*x+2) / ((x-1)^2*(x^2+x+1)). (End)
%o (Small Basic)
%o See A236326. (n = 3, i = 231).
%Y Cf. A014105 (2 center points); A234902, A234903, A234904 (3 center points); A235088, A235089 (4 center points); A236326, A236327 (5 center points).
%K nonn
%O 1,1
%A _Kival Ngaokrajang_, Jan 28 2014