A357434 a(n) is the number of distinct Q-toothpicks after the n-th stage of the structure described in A211000.

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, 15, 15, 15, 15, 15, 15, 16, 17, 18, 18, 18, 18, 19, 20, 21, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 22, 23, 24, 25, 26, 27, 28, 28

Offset: 0

Author

Paolo Xausa, Sep 28 2022

Keywords

• nonn

Comments

See A211000 for additional information.

For the definition of Q-toothpicks, see A187210.

Examples

			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).
.
n       0       1      10      15      32      39      60      65
a(n)    0       1      10      15      16      20      23      28
.
                                                                /\
                                                                \/
                                                                 \
                                                         /       /
                                                /       /\      /\
                                                \       \/      \/
              /       /\      /\      /\      /\/\    /\/\    /\/\
                        \       \       \/      \/      \/      \/
                         \      /\      /\      /\      /\      /\
                         /      \/      \/      \/      \/      \/
                        /       /\      /\      /\      /\      /\
                        \       \/      \/      \/      \/      \/
                         \      /\      /\      /\      /\      /\
                        \/      \/      \/      \/      \/      \/
.
		

Links

• N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
• Index entries for sequences related to toothpick sequences

Crossrefs

Cf. A187210, A211000, A355479.

Programs

Mathematica

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];
A357434[100]

PARI

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);
A357434(100)

Python

from sympy import isprime
def A357434(nmax):
    newtp, a, turn = ((0, 0), (1, 1)), [0, 1], 1
    tp = {newtp}
    for n in range(1, nmax):
        p1, p2 = newtp[0], newtp[1]
        if isprime(n): # Continue straight
            newtp = (p2, (2*p2[0]-p1[0], 2*p2[1]-p1[1]))
        else:          # Turn
            if n>5 and isprime(n-1): turn *= -1
            newtp = (p2, (p2[0]-turn*(p1[1]-p2[1]), p2[1]+turn*(p1[0]-p2[0])))
        tp.add(newtp)
        a.append(len(tp))
    return a[:nmax+1]
print(A357434(100))