Geodesic
On the behavior of a variant of Hofstadter's $Q$-sequence
Journal of integer sequences, Tome 10 (2007) no. 7.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: We completely solve the meta-Fibonacci recursion $V(n) = $V(n - $V(n - 1)) + $V(n - $V(n - 4))$, a variant of Hofstadter's meta-Fibonacci $Q$-sequence. For the initial conditions $V(1) = V(2) = V(3) = V(4) = 1$ we prove that the sequence $V(n)$ is monotone, with successive terms increasing by 0 or 1, so the sequence hits every positive integer. We demonstrate certain special structural properties and fascinating periodicities of the associated frequency sequence (the number of times $V(n)$ hits each positive integer) that make possible an iterative computation of $V(n)$ for any value of $n$. Further, we derive a natural partition of the $V$-sequence into blocks of consecutive terms ("$generations$") with the property that terms in one block determine the terms in the next. We conclude by examining all the other sets of four initial conditions for which this meta-Fibonacci recursion has a solution; we prove that in each case the resulting sequence is essentially the same as the one with initial conditions all ones.
Classification : 05A15, 11B37, 11B39
Keywords: meta-Fibonacci recursion, hofstadter sequence 
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     author = {Balamohan, B. and Kuznetsov, A. and Tanny, Stephen},
     title = {On the behavior of a variant of {Hofstadter's} $Q$-sequence},
     journal = {Journal of integer sequences},
     publisher = {mathdoc},
     volume = {10},
     number = {7},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2007__10_7_a4/}
}
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Balamohan, B.; Kuznetsov, A.; Tanny, Stephen. On the behavior of a variant of Hofstadter's $Q$-sequence. Journal of integer sequences, Tome 10 (2007) no. 7. http://geodesic.mathdoc.fr/item/JIS_2007__10_7_a4/