%I #24 Oct 01 2022 21:15:07

%S 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,15,15,15,15,15,15,15,15,15,15,

%T 15,15,15,15,15,15,16,17,18,18,18,18,19,20,21,22,22,22,22,22,22,22,22,

%U 22,22,22,22,22,22,22,22,22,22,22,23,24,25,26,27,28,28

%N a(n) is the number of distinct Q-toothpicks after the n-th stage of the structure described in A211000.

%C See A211000 for additional information.

%C For the definition of Q-toothpicks, see A187210.

%H 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 <a href="/index/To#toothpick">Index entries for sequences related to toothpick sequences</a>

%e In the following diagrams the A211000 structure is shown at the end of the n-th stage (Q-toothpicks are depicted as straight lines instead of circle arcs).

%e .

%e n 0 1 10 15 32 39 60 65

%e a(n) 0 1 10 15 16 20 23 28

%e .

%e /\

%e \/

%e \

%e / /

%e / /\ /\

%e \ \/ \/

%e / /\ /\ /\ /\/\ /\/\ /\/\

%e \ \ \/ \/ \/ \/

%e \ /\ /\ /\ /\ /\

%e / \/ \/ \/ \/ \/

%e / /\ /\ /\ /\ /\

%e \ \/ \/ \/ \/ \/

%e \ /\ /\ /\ /\ /\

%e \/ \/ \/ \/ \/ \/

%e .

%t A357434[nmax_]:=Module[{a={0},tp={},ep1={0,0},ep2,angle=0,turn=Pi/2},Do[If[!PrimeQ[n],If[n>5&&PrimeQ[n-1],turn*=-1];angle-=turn];tp=Union[tp,{{ep1,ep2=AngleVector[ep1,angle]}}];ep1=ep2;AppendTo[a,Length[tp]],{n,0,nmax-1}];a];

%t A357434[100]

%o (PARI)

%o A357434(nmax) = my(a=List([0,1]), newtp=[[0, 0], [1, 1]], tp=Set([newtp]), turn=1, p1, p2); if(nmax==0, return([0]));for(n=1, nmax-1, p1=newtp[1]; p2=newtp[2]; if(isprime(n), newtp=[p2, [2*p2[1]-p1[1], 2*p2[2]-p1[2]]], if(n>5 && isprime(n-1), turn*=-1); newtp=[p2, [p2[1]-turn*(p1[2]-p2[2]), p2[2]+turn*(p1[1]-p2[1])]]); tp=setunion(tp, [newtp]); listput(a,length(tp))); Vec(a);

%o A357434(100)

%o (Python)

%o from sympy import isprime

%o def A357434(nmax):

%o newtp, a, turn = ((0, 0), (1, 1)), [0, 1], 1

%o tp = {newtp}

%o for n in range(1, nmax):

%o p1, p2 = newtp[0], newtp[1]

%o if isprime(n): # Continue straight

%o newtp = (p2, (2*p2[0]-p1[0], 2*p2[1]-p1[1]))

%o else: # Turn

%o if n>5 and isprime(n-1): turn *= -1

%o newtp = (p2, (p2[0]-turn*(p1[1]-p2[1]), p2[1]+turn*(p1[0]-p2[0])))

%o tp.add(newtp)

%o a.append(len(tp))

%o return a[:nmax+1]

%o print(A357434(100))

%Y Cf. A187210, A211000, A355479.

%K nonn

%O 0,3

%A _Paolo Xausa_, Sep 28 2022