A357434 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

See A211000 for additional information.

For the definition of Q-toothpicks, see A187210.

LINKS

Table of n, a(n) for n=0..66.

N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS

Index entries for sequences related to toothpick sequences

EXAMPLE

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

/ /

/ /\ /\

\ \/ \/

/ /\ /\ /\ /\/\ /\/\ /\/\

\ \ \/ \/ \/ \/

\ /\ /\ /\ /\ /\

/ \/ \/ \/ \/ \/

/ /\ /\ /\ /\ /\

\ \/ \/ \/ \/ \/

\ /\ /\ /\ /\ /\

\/ \/ \/ \/ \/ \/

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]

PROG

(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))

CROSSREFS

Cf. A187210, A211000, A355479.

Sequence in context: A331270 A117657 A352529 * A309083 A375245 A106618

Adjacent sequences: A357431 A357432 A357433 * A357435 A357436 A357437

KEYWORD

nonn

AUTHOR

Paolo Xausa, Sep 28 2022

STATUS

approved