Geodesic
ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS
Glasgow mathematical journal, Tome 51 (2009) no. 2, pp. 243-252

Voir la notice de l'article provenant de la source Cambridge University Press

Let x0 < x1 < x2 < ⋅⋅⋅ be an increasing sequence of positive integers given by the formula xn=⌊βxn−1 + γ⌋ for n=1, 2, 3, . . ., where β > 1 and γ are real numbers and x0 is a positive integer. We describe the conditions on integers bd, . . ., b0, not all zero, and on a real number β > 1 under which the sequence of integers wn=bdxn+d + ⋅⋅⋅ + b0xn, n=0, 1, 2, . . ., is bounded by a constant independent of n. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequence qxn+1 − pxn ∈ {0, 1, . . ., q−1}, n=0, 1, 2, . . ., where x0 is a positive integer, p > q > 1 are coprime integers and xn=⌈pxn−1/q⌉ for n=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 or S:x↦(3x+1)/2) in the 3x+1 problem is also given.
DOI : 10.1017/S0017089508004655
Mots-clés : 11B50, 11B83, 11R06, 68R15 
DUBICKAS, ARTŪRAS. ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS. Glasgow mathematical journal, Tome 51 (2009) no. 2, pp. 243-252. doi: 10.1017/S0017089508004655
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