Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5

Analytic Representations of the n-anacci Constants and Generalizations Thereof


Igor Szczyrba
School of Mathematical Sciences
University of Northern Colorado
Greeley, CO 80639
USA

Rafał Szczyrba
Funiosoft, LLC
Silverthorne, CO 80498
USA

Martin Burtscher
Department of Computer Science
Texas State University
San Marcos, TX 78666
USA

Abstract:

We study generalizations of the sequence of the n-anacci constants that are constructed from the ratio limits generated by linear recurrences of an arbitrary order n with equal integer weights m. We derive the analytic representation of the class C∞ of these ratio limits and prove that, for a fixed m, the ratio limits form a strictly increasing sequence converging to m+1. We also show that the generalized n-anacci constants form a totally ordered set.

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(Concerned with sequences A000012 A000032 A000045 A000073 A000078 A000213 A000288 A000322 A001590 A001591 A001592 A001630 A001644 A002605 A007486 A010924 A015577 A020992 A021006 A023424 A026150 A028859 A028860 A030195 A057087 A057088 A057089 A057090 A057092 A057093 A066178 A073817 A074048 A074584 A077835 A079262 A080040 A081172 A083337 A084128 A085480 A086192 A086213 A086347 A094013 A100532 A100683 A104144 A104621 A105565 A105754 A105755 A106435 A106568 A108051 A108306 A116556 A121907 A122189 A122265 A123620 A123871 A123887 A124312 A125145 A134924 A141036 A141523 A145027 A164540 A164545 A164593 A168082 A168083 A168084 A170931 A181140 A214727 A214825 A214826 A214827 A214828 A214899.)

Received October 10 2014; revised version received February 23 2015. Published in Journal of Integer Sequences, May 13 2015.

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