A240828 a(1)=a(2)=0, a(3)=2; thereafter a(n) = Sum( a(n-i-s-a(n-i-1)), i=0..k-1 ), where s=0, k=3.

0, 0, 2, 2, 4, 2, 6, 4, 8, 4, 10, 6, 12, 6, 14, 8, 16, 8, 18, 10, 20, 10, 22, 12, 24, 12, 26, 14, 28, 14, 30, 16, 32, 16, 34, 18, 36, 18, 38, 20, 40, 20, 42, 22, 44, 22, 46, 24, 48, 24, 50, 26, 52, 26, 54, 28, 56, 28, 58, 30, 60, 30, 62, 32, 64, 32, 66, 34, 68, 34, 70, 36, 72, 36, 74, 38, 76, 38, 78, 40, 80, 40

Offset: 1

Author

N. J. A. Sloane, Apr 16 2014

Keywords

• nonn
• look
• hear
• easy

Comments

Is this A185048 with the leading two 1's replaced by 0's? - R. J. Mathar, Apr 17 2014.

This is true, see formulas below. - Bruno Berselli, Apr 18 2014

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..20000
• Joseph Callaghan, John J. Chew III, and Stephen M. Tanny, On the behavior of a family of meta-Fibonacci sequences, SIAM Journal on Discrete Mathematics 18.4 (2005): 794-824. See Fig. 1.4.
• Craig Knecht, Row sums of superimposed binary filled triangle.
• Index entries for Hofstadter-type sequences
• Index entries for linear recurrences with constant coefficients, signature (0,1,0,1,0,-1).

Crossrefs

Cf. A185048.

Programs

Magma

[n le 3 select 2*Floor((n-1)/2) else Self(n-Self(n-1))+Self(n-1-Self(n-2))+Self(n-2-Self(n-3)): n in [1..100]]; // Bruno Berselli, Apr 18 2014

Magma

[n-1-((-1)^n+1)*(n-(-1)^Floor(n/2)-1)/4: n in [1..80]]; // Vincenzo Librandi, Jul 12 2015

Maple

#T_s,k(n) from Callaghan et al. Eq. (1.6).
s:=0; k:=3;
a:=proc(n) option remember; global s,k;
if n <= 2 then 0
elif n = 3 then 2
else
    add(a(n-i-s-a(n-i-1)),i=0..k-1);
fi; end;
t1:=[seq(a(n),n=1..100)];

Mathematica

LinearRecurrence[{0, 1, 0, 1, 0, -1},{0, 0, 2, 2, 4, 2}, 100] (* Vincenzo Librandi, Jul 12 2015 *)

Formula

From Bruno Berselli, Apr 18 2014: (Start)

G.f.: 2*x^3*(1 + x + x^2)/((1 - x)^2*(1 + x)^2*(1 + x^2)).

a(n) = n - 1 - ((-1)^n + 1)*(n - (-1)^floor(n/2) - 1)/4. Therefore:

a(2h+1) = 2h, a(2h) = 2*floor(h/2), or also: a(4h) = a(4h+2) = 2h, a(4h+1) = 4h, a(4h+3) = 4h+2.

a(n) = a(n-2) + a(n-4) - a(n-6) for n>6. (End)