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A235089 a(n)*Pi is the total length of irregular spiral (center points: 2, 1, 3, 4) after n rotations.

%I A235089 #19 Jun 24 2022 19:58:55
%S A235089 3,10,13,20,23,30,33,40,43,50,53,60,63,70,73,80,83,90,93,100,103,110,
%T A235089 113,120,123,130,133,140,143,150,153,160,163,170,173,180,183,190,193,
%U A235089 200,203,210,213,220,223,230,233,240,243,250,253,260,263,270,273,280,283,290,293,300,303,310,313,320,323,330,333
%N A235089 a(n)*Pi is the total length of irregular spiral (center points: 2, 1, 3, 4) after n rotations.
%C A235089 Let points 2, 1, 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 non-expanded loop.
%C A235089 The alternative point order [2, 3, 1, 4] gives the same pattern with reflection, but the sequence will be 2*A047215(n). See illustration in links.
%C A235089 Conjecture: Numbers equivalent 0 or 3 modulo 10. - _Ralf Stephan_, Jan 13 2014
%H A235089 Kival Ngaokrajang, <a href="/A235089/a235089.pdf">Illustration of initial terms</a>
%F A235089 Conjecture: a(n) = -1+(-1)^n+5*n. a(n) = a(n-1)+a(n-2)-a(n-3). G.f.: x*(7*x+3) / ((x-1)^2*(x+1)). - _Colin Barker_, Jan 16 2014
%o A235089 (Small Basic)
%o A235089 a[1]=3
%o A235089 For n = 1 To 100
%o A235089 d1=3
%o A235089 If Math.Remainder(n+1,2)=0 then
%o A235089    d1=7
%o A235089 EndIf
%o A235089 a[n+1]=a[n]+d1
%o A235089 TextWindow.Write(a[n]+", ")
%o A235089 EndFor
%Y A235089 Cf. A014105*Pi (total spiral length, 2 inline center points). A234902*Pi, A234903*Pi, A234904*Pi (total spiral length, 3 inline center points).
%Y A235089 Conjectured partial sums of A010705.
%K A235089 nonn
%O A235089 1,1
%A A235089 _Kival Ngaokrajang_, Jan 03 2014