cp's OEIS Frontend

A323186 a(0) = 0, a'(0) = 0, a''(0) = 1, a''(1) = -1, a(n) = a(n-1) + a'(n), a'(n) = a'(n-1) + a''(n), a''(n) = -a''(n-1) if a(n-2) = 0, or else a''(n-1).

%I A323186 #31 Dec 19 2019 05:49:56
%S A323186 0,-1,-1,0,2,3,3,2,0,-3,-5,-6,-6,-5,-3,0,4,7,9,10,10,9,7,4,0,-5,-9,
%T A323186 -12,-14,-15,-15,-14,-12,-9,-5,0,6,11,15,18,20,21,21,20,18,15,11,6,0,
%U A323186 -7,-13,-18,-22,-25,-27,-28,-28,-27,-25,-22,-18,-13,-7,0,8,15,21,26,30,33,35,36,36,35,33,30,26,21,15,8,0,-9,-17,-24,-30,-35,-39,-42,-44,-45,-45,-44,-42,-39,-35,-30,-24,-17,-9,0
%N A323186 a(0) = 0, a'(0) = 0, a''(0) = 1, a''(1) = -1, a(n) = a(n-1) + a'(n), a'(n) = a'(n-1) + a''(n), a''(n) = -a''(n-1) if a(n-2) = 0, or else a''(n-1).
%C A323186 This sequence might be called the "Bad Driver's Sequence" as it fully "accelerates" or "decelerates" when it changes side of its "speed limit".
%H A323186 Georg Fischer, <a href="/A323186/b323186.txt">Table of n, a(n) for n = 0..1000</a>
%F A323186 a'(n) = A053615(n)*(-1)^ceiling((sqrt(4n+1)-1)/2).
%F A323186 a''(n) = (-1)^ceiling(sqrt(n)).
%F A323186 a''(n) changes sign at A002522, a(n) = 0 at A005563.
%F A323186 a(n) has local extrema (with a'(n) = 0) at the oblong numbers A002378 with the value of A000217(n)*(-1)^n, the magnitude of which is the corresponding triangular number, as such |a(n)| <= n/2.
%e A323186 a''(0) = 1, a'(0) = 0, a(0) = 0.
%e A323186 a''(1) = -1, a'(1) = 0 - 1 = -1, a(1) = 0 - 1 = -1.
%e A323186 a(2-2) = a(0) = 0, so a''(2) = -a''(1) = 1, a'(2) = -1 + 1 = 0, a(2) = -1 + 0 = -1.
%o A323186 (Haskell)
%o A323186 a(0) = 0
%o A323186 a(1) = -1
%o A323186 a(2) = -1
%o A323186 a(n) = if a(n-2) == 0 then a(n-1) + a'(n-1) - a''(n-1) else a(n-1) + a'(n-1) + a''(n-1)
%o A323186 where a'(n) = a(n) - a(n-1)
%o A323186       a''(n) = a'(n) - a'(n-1)
%o A323186 (Perl)
%o A323186 my @a = (0, -1, -1);
%o A323186 for my $n (scalar(@a)..1000) {
%o A323186     if ($a[$n - 2] == 0) {
%o A323186         $a[$n] = $a[$n - 1] + &as($n - 1) - &ass($n - 1);
%o A323186     } else {
%o A323186         $a[$n] = $a[$n - 1] + &as($n - 1) + &ass($n - 1);
%o A323186     }
%o A323186     print "$n $a[$n]\n";
%o A323186 } # for n
%o A323186 sub as  { my ($n) = @_; return $a[$n]  -  $a[$n - 1]; }
%o A323186 sub ass { my ($n) = @_; return &as($n) - &as($n - 1); }
%o A323186 # _Georg Fischer_, Feb 14 2019
%Y A323186 Cf. A000217, A000290, A002378, A002522, A005563.
%K A323186 sign,look
%O A323186 0,5
%A A323186 _Thomas Anton_, Jan 06 2019
%E A323186 a(44) corrected [18, not -18] by _Tom Duff_, Feb 14 2019