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