A188670 A quasi-periodic solution to Hofstadter's Q recurrence.

3, 6, 5, 3, 6, 8, 3, 6, 13, 3, 6, 21, 3, 6, 34, 3, 6, 55, 3, 6, 89, 3, 6, 144, 3, 6, 233, 3, 6, 377, 3, 6, 610, 3, 6, 987, 3, 6, 1597, 3, 6, 2584, 3, 6, 4181, 3, 6, 6765, 3, 6, 10946, 3, 6, 17711, 3, 6, 28657, 3, 6, 46368, 3, 6, 75025, 3, 6, 121393

Offset

0,1

Comments

a(n) is the solution to the recurrence relation a(n) = a(n-a(n-1)) +a(n-a(n-2)) [Hofstadter's Q recurrence], with the initial conditions: a(n) = 0 if n < 0; a(n) = 3 if n = 0,3; a(n) = 6 if n = 1,4; a(2) = 5; a(5) = 8.

References

F. Ruskey, Fibonacci meets Hofstadter, Fibonacci Quart. 49 (2011), no. 3, 227-230.

Links

Colin Barker, Table of n, a(n) for n = 0..1000

Nathan Fox, Linear-Recurrent Solutions to Meta-Fibonacci Recurrences, Part 1 (video), Rutgers Experimental Math Seminar, Oct 01 2015. Part 2 is vimeo.com/141111991.

F. Ruskey Fibonacci meets Hofstadter

Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,0,0,0,-1).

Formula

a(3n) = 3, a(3n+1) = 6, a(3n+2) = Fibonacci(n+5).

From Colin Barker, Nov 23 2015: (Start)

a(n) = 2*a(n-3)-a(n-9) for n>8.

G.f.: -(3*x^8+6*x^7+3*x^6+2*x^5+6*x^4+3*x^3-5*x^2-6*x-3) / ((x-1)*(x^2+x+1)*(x^6+x^3-1)).

(End)

Prog

(PARI)
A188670(n)=
{
    local(m=n%3);
    if (m==0, return(3));
    if (m==1, return(6));
    return(fibonacci(n\3+5));
}
vector(66, n, A188670(n-1)) /* show terms */ /* Joerg Arndt, Apr 08 2011 */
(PARI) Vec(-(3*x^8+6*x^7+3*x^6+2*x^5+6*x^4+3*x^3-5*x^2-6*x-3)/((x-1)*(x^2+x+1)*(x^6+x^3-1)) + O(x^100)) \\ Colin Barker, Nov 23 2015

Crossrefs

Cf. A005185 with different initial conditions.

Sequence in context: A019652 A099874 A011223 * A361985 A346602 A102621

Adjacent sequences: A188667 A188668 A188669 * A188671 A188672 A188673

Keyword

nonn,easy

Author

Frank Ruskey, Apr 08 2011

Status

approved