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A211024 Sum of all visible nodes in the structure of A211000 at n-th stage.

%I A211024 #33 Jan 16 2023 15:45:47
%S A211024 0,1,3,6,10,15,21,28,36,45,55,59,71,79,93,105,117,121,133,141,153,165,
%T A211024 177,181,193,201,209,213,217,221,237,253,285,318,350,354,358,362,400,
%U A211024 439,479,483,491,499,527,543,559,563,575,583,591,595,599,603
%N A211024 Sum of all visible nodes in the structure of A211000 at n-th stage.
%C A211024 First differs from A000217 at a(11). For n >= 13 the Q-toothpick 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.
%H A211024 Paolo Xausa, <a href="/A211024/b211024.txt">Table of n, a(n) for n = 0..9999</a> (terms 0..4999 from Carole Dubois)
%H A211024 N. J. A. Sloane, <a href="/wiki/Catalog_of_Toothpick_and_CA_Sequences_in_OEIS">Catalog of Toothpick and Cellular Automata Sequences in the OEIS</a>
%H A211024 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>
%e A211024 Consider the illustration of the nodes in structure of A211000:
%e A211024 -----------------------------------------------------
%e A211024 After 9 stages    After 10 stages    After 11 stages
%e A211024 -----------------------------------------------------
%e A211024 .
%e A211024 .    1                 1                  1
%e A211024 .  0   2             0   2              0   2
%e A211024 .        3                 3                  3
%e A211024 .          4                 4                  4
%e A211024 .        5                 5                  5
%e A211024 .      6                 6                  6
%e A211024 .        7                 7                 11
%e A211024 .          8            10   8             10   8
%e A211024 .        9                 9                  9
%e A211024 .
%e A211024 ----------------------------------------------------
%e A211024 We can see that:
%e A211024 a(9)  = 0+1+2+3+4+5+6+7+8+9 = a(8)+9 = 45
%e A211024 a(10) = 0+1+2+3+4+5+6+7+8+9+10 = a(9)+10 = 55
%e A211024 a(11) = 0+1+2+3+4+5+6+8+9+10+11 = a(10)-7+11 = 59
%t A211024 A211024[nmax_]:=Module[{ep={0,0},node=Association[],angle=3/4Pi,turn=Pi/2},Join[{0},Table[If[!PrimeQ[n],If[n>5&&PrimeQ[n-1],turn*=-1];angle-=turn];ep=AngleVector[ep,{Sqrt[2],angle}];node[ep]=n+1;Total[node],{n,0,nmax-1}]]];
%t A211024 A211024[100] (* _Paolo Xausa_, Jan 16 2023 *)
%Y A211024 Cf. A000217, A187210, A211000-A211003, A211010, A211011, A211020-A211023.
%K A211024 nonn
%O A211024 0,3
%A A211024 _Omar E. Pol_, Apr 14 2012