cp's OEIS Frontend

A360594 a(n) is the maximum number of locations 0..n-1 which can be visited in a single path starting from i=n-1, where jumps from location i to i +- a(i) are permitted (within 0..n-1) and each location can be visited up to 2 times.

%I A360594 #32 Jan 12 2024 10:05:39
%S A360594 0,2,1,2,4,3,8,1,2,2,4,2,8,5,4,6,13,14,14,13,13,16,22,3,17,16,20,13,
%T A360594 13,24,22,15,24,15,14,17,14,4,15,18,23,26,28,13,16,30,28,14,15,17,16,
%U A360594 19,16,33,18,33,32,35,39,38,40,38,39,39,36,39,38,39,41,52
%N A360594 a(n) is the maximum number of locations 0..n-1 which can be visited in a single path starting from i=n-1, where jumps from location i to i +- a(i) are permitted (within 0..n-1) and each location can be visited up to 2 times.
%C A360594 When a location is visited more than once, each such visit counts in a(n).
%e A360594 For n=0 there are no locations 0..n-1, so the sole possible path is empty which is a(0) = 0 locations.
%e A360594 For n=1, the sole possible path is location 0 twice, 0 -> 0, which is 2 locations a(1) = 2.
%e A360594 For n=12, a path of a(12) = 8 locations visited starting from n-1 = 11 is:
%e A360594   11 -> 9 -> 11 -> 9 -> 7 -> 8 -> 10 -> 6
%e A360594 Locations 11 and 9 are visited twice each and the others once.
%e A360594   i    = 0  1  2  3  4  5  6  7  8  9 10 11
%e A360594   a(i) = 0, 2, 1, 2, 4, 3, 8, 1, 2, 2, 4, 2
%e A360594                                     2<----2
%e A360594                                      ---->2
%e A360594                               1<----2<----
%e A360594                                ->2---->4
%e A360594                            8<----------
%o A360594 (Python)
%o A360594 def A(lastn,times=2,mode=0):
%o A360594   a,n=[0],0
%o A360594   while n<lastn:
%o A360594     d,i,v,o,g,r=[[n]],0,1,[],0,0
%o A360594     while len(d)>0:
%o A360594       if len(d[-1])>v: v,o=len(d[-1]),d[-1][:]
%o A360594       if d[-1][-1]-a[d[-1][-1]]>=0:
%o A360594         if d[-1].count(d[-1][-1]-a[d[-1][-1]])<times:g=1
%o A360594       if d[-1][-1]+a[d[-1][-1]]<=n:
%o A360594         if d[-1].count(d[-1][-1]+a[d[-1][-1]])<times:
%o A360594           if g>0: d.append(d[-1][:])
%o A360594           d[-1].append(d[-1][-1]+a[d[-1][-1]])
%o A360594           r=1
%o A360594       if g>0:
%o A360594         if r>0: d[-2].append(d[-2][-1]-a[d[-2][-1]])
%o A360594         else: d[-1].append(d[-1][-1]-a[d[-1][-1]])
%o A360594         r=1
%o A360594       if r==0: d.pop()
%o A360594       r,g=0,0
%o A360594     a.append(v)
%o A360594     n+=1
%o A360594     if mode==0: print(n,a[n])
%o A360594     if mode>0:
%o A360594       u,q=0,[]
%o A360594       while u<len(o):
%o A360594         q.append(a[o[u]])
%o A360594         u+=1
%o A360594       print(n,a[n],q,o)
%o A360594   return a
%Y A360594 Cf. A360593, A360595.
%K A360594 nonn
%O A360594 0,2
%A A360594 _S. Brunner_, Feb 13 2023