Pinn, Klaus A chaotic cousin of Conway’s recursive sequence. (English) Zbl 0959.05007 Exp. Math. 9, No. 1, 55-66 (2000). Summary: I introduce the recurrence \(D(n)= D(D(n-1))+ D(n- 1- D(n- 2))\), \(D(1)= D(2)= 1\), and study it by means of computer experiments. The definition of \(D(n)\) has some similarity to that of Conway’s sequence defined by \(a(n)= a(a(n- 1))+ a(n- a(n- 1))\), \(a(1)= a(2)= 1\). However, unlike the completely regular and predictable behaviour of \(a(n)\), the \(D\)-numbers exhibit chaotic patterns. In its statistical properties, the \(D\)-sequence shows striking similarities with Hofstadter’s \(Q(n)\)-sequence, given by \(Q(n)= Q(n- Q(n- 1))+ Q(n- Q(n- 2))\), \(Q(1)= Q(2)= 1\); see Douglas R. Hofstadter [Gödel, Escher, Bach: an eternal golden braid (1979; reprint 1981; Zbl 0457.03001)]. Compared to the Hofstadter sequence, \(D\) shows higher structural order. It is organized in well-defined “generations”, separated by smooth and predictable regions. The article is complemented by a study of two further recurrence relations with definitions similar to those of the \(Q\)-numbers. There is some evidence that the different sequences studied share a universality class. Cited in 7 Documents MSC: 05A15 Exact enumeration problems, generating functions 11B83 Special sequences and polynomials 11B37 Recurrences Keywords:Conway’s sequence; \(D\)-sequence; Hofstadter’s \(Q(n)\)-sequence; \(Q\)-numbers Citations:Zbl 0457.03001 × Cite Format Result Cite Review PDF Full Text: DOI arXiv EuDML Online Encyclopedia of Integer Sequences: Hofstadter-Conway \(10000 sequence: a(n) = a(a(n-1)) + a(n-a(n-1)) with a(1) = a(2) = 1\) A chaotic cousin of the Hofstadter-Conway sequence A004001. a(n) = sum_(1..n) [S2(n)mod 2 - floor(5*S2(n)/7)mod 2], where S2(n) = binary weight of n. Conway variant a(n) = 1+a(n-1-(n mod a(n-1))), with a(1)=1. Sequence defined by the recursion a(n)=(1+a(n-1-(n mod a(n-1)))-(-1)^n*a(n-1)) mod n, with a(1)=0. Sequence defined by the recursion a(n) = (1/2)*((1-signum(abs(b(n))-n))*b(n)+(1+signum(abs(b(n))-n))*a(n-1)), with a(1)=1 and b(n)=1+a(n-1-(n mod a(n-1)))-(-1)^n*a(n-1). a(n) = a(n-1-a(n-1)) + a(n-a(n-2)) for n>2; starting with a(1) = a(2) = 1. References: [1] Barbeau E., Electron. J. Combin. 3 (1) pp 9– (1996) [2] Barbeau E. J., Electron. J. Combin. 4 (1) pp 11– (1997) [3] Bouchaud J.-P., Phys. Rep. 195 (4) pp 127– (1990) · doi:10.1016/0370-1573(90)90099-N [4] Conolly B. W., Fibonacci & Lucas numbers, and the golden section: theory and applications pp 127– (1989) [5] Guy, R. K. 1981.Unsolved problems in number theory186–190. New York: Springer. [Guy 1981], Problem Books in Math., See also his column in Amer. Math. Monthly 93 (1986) [6] Hofstadter D. R., Gödel, Escher, Bach: an eternal golden braid (1979) [7] Hofstadter, D. R. September 2 1988. September 2, [Hofstadter 1988], Letters to C. Mallows and J. H. Conway [8] Kubo T., Discrete Math. 152 (1) pp 225– (1996) · Zbl 0852.05010 · doi:10.1016/0012-365X(94)00303-Z [9] Mallows C. L., Amer. Math. Monthly 98 (1) pp 5– (1991) · Zbl 0738.11014 · doi:10.2307/2324028 [10] Pinn K., Complexity 4 (3) pp 41– (1999) · doi:10.1002/(SICI)1099-0526(199901/02)4:3<41::AID-CPLX8>3.0.CO;2-3 [11] Tanny S. M., Discrete Math. 105 (1) pp 227– (1992) · Zbl 0766.11008 · doi:10.1016/0012-365X(92)90145-6 [12] Yao, A. K. 1997. [Yao 1997], Private communication via D. R. Hofstadter This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.