OFFSET
0,5
COMMENTS
It appears there is an infinite family of this type of curves or structures in which the terms of a sequence of positive integers are represented as inflection points and the gaps between them are essentially represented as nodes of spirals. For example: consider a structure formed by Q-toothpicks of size = Axxxxxa connected by their endpoints in which the inflection points are the exposed endpoints at stage Axxxxxb(n), where both Axxxxxa and Axxxxxb are sequences with positive integers. Also instead of Q-toothpicks we can use semicircumferences or also 3/4 of circumferences. For the definition of Q-toothpicks see A187210.
We start at stage 0 with no Q-toothpicks.
At stage 1 we place a Q-toothpick of size 1 centered at (1,0) with its endpoints at (0,0) and (1,1). Since 1 is a positive triangular number we have that the end of the curve is also an inflection point.
At stage 2 we place a Q-toothpick of size 2 centered at (1,3) with its endpoints at (1,1) and (3,3).
At stage 3 we place a Q-toothpick of size 3 centered at (0,3) with its endpoints at (3,3) and (0,6). Since 3 is a positive triangular number we have that the end of the curve is also an inflection point.
At stage 4 we place a Q-toothpick of size 4 centered at (0,10) with its endpoints at (0,6) and (-4,10).
And so on...
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..9999
N. J. A. Sloane, Catalog of Toothpick and Cellular Automata Sequences in the OEIS
Paolo Xausa, Animation of terms n = 0..41 (first 21 coordinate pairs), where orange dots are toothpick endpoints (hollow dots are inflection points) and blue dots are toothpick centers
Paolo Xausa, Scatterplot of 10^5 coordinate pairs
EXAMPLE
-------------------------------------
Stage n also The end as
the size of Pair inflection
Q-toothpick (x y) point
-------------------------------------
. 0 0, 0, -
. 1 1, 1, Yes
. 2 3, 3, -
. 3 0, 6, Yes
. 4 -4, 10, -
. 5 1, 15, -
. 6 7, 9, Yes
. 7 14, 2, -
. 8 22, 10, -
. 9 13, 19, -
. 10 3, 9, Yes
. 11 -8, -2, -
. 12 -20, 10, -
. 13 -7, 23, -
. 14 7, 9, -
. 15 -8, -6, Yes
MATHEMATICA
A210838[nmax_]:=Module[{ep={0, 0}, angle=3/4Pi, turn=Pi/2, infl=0}, Join[{ep}, Table[If[n>1&&IntegerQ[Sqrt[8(n-1)+1]], infl++, If[Mod[infl, 2]==1, turn*=-1]; angle-=turn; infl=0]; ep=AngleVector[ep, {Sqrt[2]n, angle}], {n, nmax}]]];
A210838[100] (* Generates 101 coordinate pairs *) (* Paolo Xausa, Jan 12 2023 *)
PROG
(PARI)
A210838(nmax) = my(ep=vector(nmax+1), turn=1, infl=0, ep1, ep2); ep[1]=[0, 0]; if(nmax==0, return(ep)); ep[2]=[1, 1]; for(n=2, nmax, ep1=ep[n-1]; ep2=ep[n]; if(issquare((n-1)<<3+1), infl++; ep[n+1]=[ep2[1]+n*sign(ep2[1]-ep1[1]), ep2[2]+n*sign(ep2[2]-ep1[2])], if(infl%2, turn*=-1); infl=0; ep[n+1]=[ep2[1]-turn*n*sign(ep1[2]-ep2[2]), ep2[2]+turn*n*sign(ep1[1]-ep2[1])])); ep;
A210838(100) \\ Generates 101 coordinate pairs - Paolo Xausa, Jan 12 2023
(Python)
from numpy import sign
from sympy import integer_nthroot
def A210838(nmax):
ep, turn, infl = [(0, 0), (1, 1)], 1, 0
for n in range(2, nmax + 1):
ep1, ep2 = ep[-2], ep[-1]
if integer_nthroot(((n - 1) << 3) + 1, 2)[1]: # Continue straight
infl += 1
dx = n * sign(ep2[0] - ep1[0])
dy = n * sign(ep2[1] - ep1[1])
else: # Turn
if infl % 2: turn *= -1
infl = 0
dx = turn * n * sign(ep2[1] - ep1[1])
dy = turn * n * sign(ep1[0] - ep2[0])
ep.append((ep2[0] + dx, ep2[1] + dy))
return ep[:nmax+1]
print(A210838(100)) # Generates 101 coordinate pairs - Paolo Xausa, Jan 12 2023
CROSSREFS
AUTHOR
Omar E. Pol, Mar 28 2012
EXTENSIONS
a(30)-a(33) corrected and more terms by Paolo Xausa, Jan 12 2023
STATUS
approved