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

%I A235088 #19 Oct 22 2021 15:31:42
%S A235088 3,6,17,28,47,66,93,120,155,190,233,276,327,378,437,496,563,630,705,
%T A235088 780,863,946,1037,1128,1227,1326,1433,1540,1655,1770,1893,2016,2147,
%U A235088 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
%N A235088 a(n)*Pi is the total length of irregular spiral (center points: 1, 2, 3, 4) after n rotations.
%C A235088 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.
%H A235088 Kival Ngaokrajang, <a href="/A235088/a235088.pdf">Illustration of initial terms</a>
%F A235088 a(n) = 2*floor((n-1)^2/4) + 3*ceiling(n^2/2) (conjectured). - _Ralf Stephan_, Jan 13 2014
%F A235088 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
%o A235088 (Small Basic)
%o A235088 a[1]=3
%o A235088 d1=3
%o A235088 For n = 1 To 100
%o A235088   If Math.Remainder(n+3,2)=1 then
%o A235088     d1=d1+8
%o A235088   EndIf
%o A235088   a[n+1]=a[n]+d1
%o A235088   TextWindow.Write(a[n]+", ")
%o A235088 EndFor
%Y A235088 Cf. A014105*Pi (total spiral length, 2 inline center points). A234902*Pi, A234903*Pi, A234904*Pi (total spiral length, 3 inline center points).
%K A235088 nonn
%O A235088 1,1
%A A235088 _Kival Ngaokrajang_, Jan 03 2014