Journal of Integer Sequences, Vol. 19 (2016), Article 16.3.8

Geometric 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 introduce geometric representations of the sequence of the n-anacci constants and generalizations thereof that consist of the ratio limits generated by linear recurrences of an arbitrary order n with equal real weights p > 0. We represent the n-anacci constants and their generalizations geometrically by means of the dilation factors of dilations transforming collections of compact convex sets with increasing dimensions n.

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(Concerned with sequences A000073 A000078 A000213 A000288 A000322 A000383 A001590 A001591 A001592 A001630 A001631 A001644 A001648 A002605 A003691 A007486 A010924 A020992 A021006 A023424 A026150 A028831 A028859 A028860 A030195 A052945 A057087 A057088 A057089 A057090 A057091 A057092 A057093 A060455 A066178 A073728 A073817 A074048 A074584 A077835 A077836 A077843 A079262 A080040 A081172 A083337 A084128 A085480 A086192 A086213 A086347 A093175 A094013 A100532 A100683 A101757 A101758 A102026 A103771 A104144 A104621 A105754 A105755 A106435 A106565 A106568 A108051 A108306 A116556 A119826 A121907 A122189 A122265 A122997 A123526 A123620 A123871 A123887 A124312 A124313 A125145 A127193 A127194 A127624 A134927 A135055 A135056 A135491 A135492 A141036 A141523 A145027 A145028 A145029 A145030 A155116 A155127 A155144 A160175 A162562 A163551 A164540 A164545 A164593 A164603 A164607 A168082 A168083 A168084 A170931 A172011 A172012 A180033 A180167 A181140 A186830 A188714 A189737 A189743 A189749 A202206 A207539 A213665 A214727 A214825 A214826 A214827 A214828 A214829 A214830 A214831 A214899 A220469 A220493 A228603 A248959 A249169 A251653 A251654 A251655 A251656 A251672 A251703 A251704 A251705 A251706 A251707 A251708 A251709 A251710 A251711 A251712 A251713 A251714 A251740 A251741 A251742 A251744 A251745 A251746 A251747 A251748 A251749 A251750 A251751 A251752 A251759 A251760 A251761 A251762 A251763 A251764 A251765 A251766.)

Received December 5 2015; revised version received March 8 2016. Published in Journal of Integer Sequences, April 7 2016.

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