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.
For n = 3: a(3) != 1 or else midpoint((1, 1), (3, 1)) = midpoint((2, 1), (2, 1)), so a(3) = 2. For n = 5: a(5) != 1 or else midpoint((1, 1), (5, 1)) = midpoint((2, 1), (4, 1)); a(5) != 2 or else midpoint((2, 1), (5, 2)) = midpoint((3, 2), (4, 1)); a(5) != 3 or else midpoint((1, 1), (5, 3)) = midpoint((3, 2), (3, 2)); so a(5) = 4.
a=[];for(n=1,190,a=concat(a,1);while(hasAP(a,4),a[#a]++));a \\ See A248625 for hasAP().
cpdef FourFree(int n):
cdef int i, r, k, s, L1, L2, L3
cdef list L, Lb
cdef set b
L=[1, 1, 1]
for k in range(3, n):
b=set()
for i in range(k):
if 3*((k-i)/3)==k-i:
r=(k-i)/3
L1, L2, L3=L[i], L[i+r], L[i+2*r]
s=3*(L2-L1)+L1
if s>0 and L3==2*(L2-L1)+L1:
b.add(s)
if 1 not in b:
L.append(1)
else:
Lb=list(b)
Lb.sort()
for t in Lb:
if t+1 not in b:
L.append(t+1)
break
return L
# Sébastien Palcoux, Aug 28 2019
For n = 4: - the first 4 terms of the sequence are: 0, 1, 1, 2, - a(4) cannot equal 0 due to the progression b(-4) = 0, b(0) = 0, b(4) = 0, - a(4) cannot equal 1 due to the progression b(-2) = 1, b(1) = 1, b(4) = 1, - a(4) cannot equal 2 due to the progression b(0) = 0, b(2) = 1, b(4) = 2, - a(4) cannot equal 3 due to the progression b(2) = 1, b(3) = 2, b(4) = 3, - we chose a(4) = 4 as this does not induce arithmetic progressions.
a=[];for(n=1,190,a=concat(a,0);while(hasAP(a,4),a[#a]++));a \\ See A248625 for hasAP().
a=[];for(n=1,190,a=concat(a,0);while(hasAP(a,5),a[#a]++));a \\ See A248625 for hasAP(). Use concat(a,1) for the "positive integer" variant.
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