Journal of Integer Sequences, Vol. 19 (2016), Article 16.7.6

Hyperfibonacci Sequences and Polytopic Numbers

Ligia L. Cristea
Karl-Franzens-Universität Graz
Institute for Mathematics and Scientific Computing
Heinrichstrasse 36
A-8010 Graz

Ivica Martinjak
University of Zagreb Faculty of Science
Bijenička 32
HR-10000 Zagreb

Igor Urbiha
Zagreb University of Applied Sciences (former Polytechnic of Zagreb)
Vrbik 8
HR-10000 Zagreb


We prove that the difference between the nth hyperfibonacci number of the rth generation and its two consecutive predecessors is the nth regular (r-1)-topic number. Using this fact, we provide an equivalent recursive definition of the hyperfibonacci sequences, and derive an extension of the Binet formula. We also prove further identities involving both hyperfibonacci and hyperlucas sequences, in full generality.

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(Concerned with sequences A000045 A000071 A001924 A014162 A014166.)

Received July 25 2015; revised versions received May 11 2016; September 1 2016. Published in Journal of Integer Sequences, September 7 2016.

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