AnRsin2n

AnRsin2n is a discrete-time mathematical sequence defined by

an=R⋅sin(2n) for n∈ℤ, where R is a real constant representing the amplitude of the oscillation. The argument of the sine function, 2n, is measured in radians, so each successive element of the sequence corresponds to a phase increment of 2 rad. Because the sine function has a fundamental period of 2π, the sequence itself does not possess a simple integer period; there is no positive integer k for which sin(2(n+k))=sin(2n) for all n. Consequently, AnRsin2n is quasi‑periodic, exhibiting a continuous range of values without repeating exactly over integer steps.

The sequence is bounded between –R and +R, attaining its extrema when sin(2n)=±1. These extrema occur whenever 2n=π/2+2πm or 3π/2+2πm for integers m, which rarely align with integer values of n because π is irrational. Therefore the sequence never reaches its maximal amplitude at integer indices except in special cases when R=0. The average value of the sequence over a long interval is zero due to the symmetric nature of the sine function.

AnRsin2n frequently appears in the analysis of digital signals and oscillators. In signal processing, it models a pure sinusoid sampled at a rate that yields a phase increment of 2 rad per sample. Its Fourier representation consists of two complex exponentials at frequencies ±1, scaled by R/2i, making it relevant for spectral analysis of discrete-time systems. In control theory and discrete dynamical systems, AnRsin2n can serve as a forcing term or as part of a solution to linear difference equations. Its simplicity also makes it a common example when illustrating properties of bounded, non‑periodic sequences in introductory courses on sequences and series.