This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
From the starting point (0,0) labeled a(0) = 0, the king can reach the point (1,1) labeled 2, which is the smallest possible prime number, so a(1) = 2. Then the king can reach (1,0) labeled 3 which is the next smaller prime number, so a(2) = 3. From there it can go to (-1,0) labeled 5 = a(3), and then to (0,-1) labeled a(4) 7 = a(4). From there, the only available prime number is 23 = a(5) at (1,-2). The king continues in that south-east direction, before walking in north-east direction and then back and further south-east up to and beyond the point (10,-10). Then it goes back in the opposite north-west direction up to (-8,7) and (-5, 8), before heading to the point (3,0) where it gets stuck.
from sympy import isprime # = A010051 def square_number(z): return int(4*y**2-y-x if (y := z.imag) >= abs(x := z.real) else 4*x**2-x-y if -x>=abs(y) else (4*y-3)*y+x if -y>=abs(x) else (4*x-3)*x+y) def A383183(n, moves=(1, 1+1j, 1j, 1j-1, -1, -1-1j, -1j, 1-1j)): if not hasattr(A:=A383183, 'terms'): A.terms=[0]; A.pos=0; A.path=[0] while len(A.terms) <= n: try: _,s,z = min((1-isprime(s), s, z) for d in moves if (s := square_number(z := A.pos+d)) not in A.terms) except ValueError: raise IndexError(f"Sequence has only {len(A.terms)} terms") A.terms.append(s); A.pos = z; A.path.append(z) return A.terms[n] A383183(999) # gives IndexError: Sequence has only 172 terms A383183.terms # shows the full sequence; append [:N] to show only N terms import matplotlib.pyplot as plt # this and following to plot the path: plt.plot([z.real for z in A383183.path], [z.imag for z in A383183.path]) plt.show()
For n = 2, a(2) = 4 because moving to 2 or 3 does not pass through a wall.
def square_number(z): return int(4*y**2-y-x if (y := z.imag) >= abs(x := z.real)
else 4*x**2-x-y if -x>=abs(y) else (4*y-3)*y+x if -y>=abs(x) else (4*x-3)*x+y)
def A375925(n):
if not hasattr(A:=A375925, 'terms'): A.terms=[1]; A.pos=0
while len(A.terms) < n:
s,d = min((s,d) for d in (1, 1+1j, 1j, 1j-1, -1, -1-1j, -1j, 1-1j) if
abs((s:=1+square_number(A.pos+d))-A.terms[-1]) > 2 and s not in A.terms)
A.terms.append(s); A.pos += d
return A.terms[n-1] # M. F. Hasler, May 07 2025
def diamond_number(z):
x, y = int(z.real), int(z.imag); d = abs(x)+abs(y)
return 2*d*(d-1)+((x if y<0 else d+y)if x>0 else 2*d-x if y>0 else 3*d-y)
def A383187(n):
if not hasattr(A:=A383187, 'terms'): A.terms=[0]; A.pos=0; # A.track=[0]
while len(A.terms) <= n:
s, d = min((s, d) for d in (1, 1+1j, 1j, 1j-1, -1, -1-1j, -1j, 1-1j) if
abs((s:=diamond_number(A.pos+d))-A.terms[-1]) > 1 and s not in A.terms)
A.terms.append(s); A.pos += d; # A.track.append(A.pos)
return A.terms[n]
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