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).

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, -12, -14, -15, -15, -14, -12, -9, -5, 0, 6, 11, 15, 18, 20, 21, 21, 20, 18, 15, 11, 6, 0, -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

Offset

0,5

Comments

This sequence might be called the "Bad Driver's Sequence" as it fully "accelerates" or "decelerates" when it changes side of its "speed limit".

Links

Georg Fischer, Table of n, a(n) for n = 0..1000

Formula

a'(n) = A053615(n)*(-1)^ceiling((sqrt(4n+1)-1)/2).

a''(n) = (-1)^ceiling(sqrt(n)).

a''(n) changes sign at A002522, a(n) = 0 at A005563.

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.

Example

a''(0) = 1, a'(0) = 0, a(0) = 0.
a''(1) = -1, a'(1) = 0 - 1 = -1, a(1) = 0 - 1 = -1.
a(2-2) = a(0) = 0, so a''(2) = -a''(1) = 1, a'(2) = -1 + 1 = 0, a(2) = -1 + 0 = -1.

Prog

(Haskell)
a(0) = 0
a(1) = -1
a(2) = -1
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)
where a'(n) = a(n) - a(n-1)
      a''(n) = a'(n) - a'(n-1)
(Perl)
my @a = (0, -1, -1);
for my $n (scalar(@a)..1000) {
    if ($a[$n - 2] == 0) {
        $a[$n] = $a[$n - 1] + &as($n - 1) - &ass($n - 1);
    } else {
        $a[$n] = $a[$n - 1] + &as($n - 1) + &ass($n - 1);
    }
    print "$n $a[$n]\n";
} # for n
sub as  { my ($n) = @_; return $a[$n]  -  $a[$n - 1]; }
sub ass { my ($n) = @_; return &as($n) - &as($n - 1); }
# Georg Fischer, Feb 14 2019

Crossrefs

Cf. A000217, A000290, A002378, A002522, A005563.

Sequence in context: A203400 A077869 A076585 * A022906 A285408 A101914

Adjacent sequences: A323183 A323184 A323185 * A323187 A323188 A323189

Keyword

sign,look

Author

Thomas Anton, Jan 06 2019

Extensions

a(44) corrected [18, not -18] by Tom Duff, Feb 14 2019

Status

approved