preissmann8. 167KB Jun 04 2011 12:09:22 AM

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Article 05.3.5

Journal of Integer Sequences, Vol. 8 (2005), 2

3 6 1 47

A Self-Indexed Sequence

Emmanuel Preissmann

11 rue Vinet

1004 Lausanne

Switzerland

emmanuel.preissmann@lth.epfl.ch

Abstract

We investigate the integer sequence (tn)_n∈Z defined by tn = 0 if n ≤ 0, t1 = 1, and tn = Pni=1−1tn−ti for n ≥ 2. This sequence has the following properties: if we

considerfn(X) :=−1 +Pn_i=1Xti _{and takex}

nto be the real positive number such that

fn(xn) = 0, then

lim n→∞

tn

tn+1

= lim

n→∞xn= 0.410098516· · ·

Moreover, if u is the real positive number such that 1 = P∞_i=1u−ti_{, then there is a}

positive constantM such that tn∼M un.

1

Definitions and main results

If we look in theOnline Encyclopedia of Integer Sequences of N. J. A. Sloane [1], we find a remarkable sequence by (tn)_n∈Z defined by tn= 0 if n ≤0,t1 = 1 and

tn = n−1

X

i=1

tn−ti (1.1)

for n ≥2. This sequence is due to Robert Lozyniak (A052109 in Sloane) and is a cousin of the Hofstadter-Conway $10,000 challenge sequence (A004001in Sloane).

We find t₂ = 1, t₃ = 2, t₄ = 5, t₅ = 12, t₆ = 30, t₇ = 73, . . . Observe that if n ≥ 2 ,

t_n+1 = 2tn+· · · ≥2tn. Since t4 = 4 + 1, we have

tn ≥n+ 1 (n≥4) (1.2)

The serieP∞

i=1Xt

i _{= 2X+X}2_{+· · · converges for|X|<1. Letube the real positive number}

such that _∞

X

i=1

We easily see that 2< u <3. Indeed we have P∞

Theorem 1.1. There exists a positive constant M such that

lim

n=1 is strictly decreasing. Indeed,

n

This proves thatx_n+1 < xn for every n. Moreover, this sequence is bounded, so it converges

to an element, say v. The function f∞(x) = −1 +

if n is big enough. Finally,

f∞(v) = lim

n→∞f∞(xn) = 0.

That is,v = 1_u, because _u1 is the only positive zero of f∞.

2

Proof of the Theorem

We begin with a lemma:

Proof. Without loss of generality, we can suppose that vn ≤ 1₂. If an infinite number of vn > 1₂, the claims 2.1 and 2.2 are both wrong, otherwise we can take out the finite number

of vn > 1₂. By Taylor expansion, we have for 0≤x≤ 1₂

So the lemma is proved.

Lemma 2.2. There exists a constant d such that

0< d≤ tn positive integer n.

Proof. If n ≤ 1, this is evident. Suppose that n ≥ 1 and by induction that ti ≤ ui−1 when

By induction we have

∞

By the definition ofCN

Proof. Suppose that d′

Proof of the Theorem. According to Lemmas (2.4), (2.5), (2.6) and the definition of CN,

we have, for 1≤N < n and k ≥0:

tn+k ≤CNun+k−t′k+2(CNun−tn) +akuN ≤CNun+k−d′uk+2(CNun−tn) +uN+k

Letǫ > 0. There exists N such that C ≤CN < C+ǫ, and n > N such that uN−n < ǫ and tn

un < D+ǫ.In these conditions, we have

tn+k

un+k < C +ǫ−d

′

u2(C−D−ǫ) +ǫ.

There exists k such that tn+k

un+k > C−ǫ. Then C−ǫ < C+ǫ−d

′_u2_{(C−D−ǫ) +ǫ. Letting} ǫ tend to 0 gives C ≤C−d′_u2_{(C−D). Hence d}′_u2_{(C−D)≤0. This implies C ≤D and}

thusC =D.

3

Acknowledgments

I am grateful to M. Mischler for valuable help and many fruitful discussions during the course of this work.

References

[1] N. J. A. Sloane, Online Encyclopedia of Integer Sequences,

http://www.research.att.com/˜njas/sequences/

2000 Mathematics Subject Classification: Primary 11Y55.

Keywords: integer sequences.

(Concerned with sequence A052109.)

Received May 16 2005; revised version received July 28 2005. Published inJournal of Integer Sequences, August 2 2005.