Abstract
Linear recurrences with constant coefficients are an interesting class of recurrence equations that can be solved explicitly. The most famous example are certainly the Fibonacci numbers with the equation f(n) = f(n−1) + f(n−2) and the quite non-obvious closed form <table class="display dcenter"><tr style="vertical-align:middle"><td class="dcell"><table class="display"><tr><td class="dcell" style="text-align:center">1</td></tr><tr><td class="hbar"></td></tr><tr><td class="dcell" style="text-align:center"><table class="display"><tr style="vertical-align:middle"><td class="dcell">√</td><td class="dcell"><table style="border:0;border-spacing:1;border-collapse:separate;" class="cellpadding0"><tr><td class="hbar"></td></tr><tr><td style="text-align:center;white-space:nowrap" >5</td></tr></table></td></tr></table></td></tr></table></td><td class="dcell"> (ϕn − (−ϕ)−n)</td></tr></table> where ϕ is the golden ratio.
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