cp's OEIS Frontend

A211011 Value on the axis "y" of the endpoint of the structure (or curve) of A211000 at n-th stage.

%I A211011 #66 Jan 14 2023 22:00:10
%S A211011 0,1,0,-1,-2,-3,-4,-5,-6,-7,-6,-5,-4,-3,-2,-1,-2,-3,-4,-5,-6,-7,-6,-5,
%T A211011 -4,-3,-4,-5,-4,-3,-2,-1,0,1,0,-1,0,1,2,3,2,1,0,-1,-2,-3,-2,-1,0,1,0,
%U A211011 -1,0,1,2,3,2,1,2,3,4,5,6,7,6,5,6,7,8,9,8,7,6,5
%N A211011 Value on the axis "y" of the endpoint of the structure (or curve) of A211000 at n-th stage.
%C A211011 For n >= 13 the structure of A211000 looks like essentially a column of tangent circles of radius 1. The structure arises from the prime numbers A000040. The behavior seems to be as modular arithmetic but in a growing structure. Note that all odd numbers > 1 are located on the main axis of the structure. For the number of circles after n-th stage see A211020. For the values on the axis "x" see A211010. For the values for the n-th prime see A211023.
%H A211011 Paolo Xausa, <a href="/A211011/b211011.txt">Table of n, a(n) for n = 0..9999</a>
%H A211011 N. J. A. Sloane, <a href="http://oeis.org/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H A211011 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%F A211011 abs(a(n)-a(n+1)) = 1.
%e A211011 Consider the illustration of the structure of A211000:
%e A211011 ------------------------------------------------------
%e A211011 .           After           After            After
%e A211011 .  y      9 stages        10 stages        11 stages
%e A211011 ------------------------------------------------------
%e A211011 .  2
%e A211011 .  1        1               1                1
%e A211011 .  0      0   2           0   2            0   2
%e A211011 . -1            3               3                3
%e A211011 . -2              4               4                4
%e A211011 . -3            5               5                5
%e A211011 . -4          6               6                6
%e A211011 . -5            7               7               11
%e A211011 . -6              8          10   8           10   8
%e A211011 . -7            9               9                9
%e A211011 . -8
%e A211011 We can see that a(7) = a(11) = -5.
%t A211011 A211011[nmax_]:=Module[{ep={0,0},angle=3/4Pi,turn=Pi/2},Join[{0},Table[If[!PrimeQ[n],If[n>5&&PrimeQ[n-1],turn*=-1];angle-=turn];Last[ep=AngleVector[ep,{Sqrt[2],angle}]],{n,0,nmax-1}]]];
%t A211011 A211011[100] (* _Paolo Xausa_, Jan 14 2023 *)
%Y A211011 Bisection of A211000.
%Y A211011 Cf. A187210, A210838, A210841, A211001-A211003, A211010, A211020-A211024.
%K A211011 sign,look
%O A211011 0,5
%A A211011 _Omar E. Pol_, Mar 30 2012