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Results of Terras and Everett imply that, given any $\\epsilon>0$, almost all $m\\in\\mathbb{Z}^+$ (in the sense of natural density) fulfill $(\\frac{\\sqrt{3}}{2})^km^{1-\\epsilon}\\leq \\mathsf{T}^k(m)\\leq (\\frac{\\sqrt{3}}{2})^km^{1+\\epsilon}$ simultaneously for all $0\\leq k\\leq \\alpha\\log m$ with $\\alpha=(\\log 2)^{-1}\\approx 1.443$. We extend this result to $\\alpha=2(\\log\\frac{4}{3})^{-1}\\approx 6.952$, which is the maximally possible value. 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Results of Terras and Everett imply that, given any $\\epsilon>0$, almost all $m\\in\\mathbb{Z}^+$ (in the sense of natural density) fulfill $(\\frac{\\sqrt{3}}{2})^km^{1-\\epsilon}\\leq \\mathsf{T}^k(m)\\leq (\\frac{\\sqrt{3}}{2})^km^{1+\\epsilon}$ simultaneously for all $0\\leq k\\leq \\alpha\\log m$ with $\\alpha=(\\log 2)^{-1}\\approx 1.443$. We extend this result to $\\alpha=2(\\log\\frac{4}{3})^{-1}\\approx 6.952$, which is the maximally possible value. 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