# Odd and even repetition sequences of independent domination number

Leomarich F. Casinillo
Notes on Number Theory and Discrete Mathematics
Print ISSN 1310–5132, Online ISSN 2367–8275
Volume 26, 2020, Number 1, Pages 8—20
DOI: 10.7546/nntdm.2020.26.1.8-20

## Details

### Authors and affiliations

Leomarich F. Casinillo
Department of Mathematics and Physics, Visayas State University
Visca, Baybay City, Leyte, Philippines

### Abstract

Let {Pn}n=1 be a sequence of paths. The odd repetition sequence denoted by {𝜌𝑘𝑜:𝑘∈ℕ} is a sequence of natural numbers in which odd numbers are repeated once and defined by {𝜌𝑘𝑜}={1,1,2,3,3,4,5,5,…}={𝑖(𝑃𝑛)} where 𝑛=2𝑘−1. The even repetition sequence denoted by {𝜌𝑘𝑒:𝑘∈ℕ} is a sequence of natural numbers, in which even numbers are repeated once and defined by {𝜌𝑘𝑒}={1,2,2,3,4,4,5,6,6,…}={𝑖(𝑃𝑛)}, where 𝑛 = 2𝑘. In this paper, the explicit formula that shows the values of the element of two sequences {𝜌𝑘𝑜}} and {𝜌𝑘𝑒} that depends on the subscript 𝑘 were constructed. Also, the formula that relates the partial sum of the elements of the said sequences, which depends on the subscript 𝑘 and order of the sequence of paths, were established. Further, the independent domination number of the triangular grid graph 𝑇𝑚 = (𝑉(𝑇𝑚), 𝐸(𝑇𝑚)) will be determined using the said sequences and the two sequences will be evaluated in relation to the Fibonacci sequence {𝐹𝑛} along with the order of the path.

### Keywords

• Odd repetition sequence
• Even repetition sequence
• Independent domination number
• Fibonacci numbers
• Triangular grid graph

• 05C69

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