Alternative Recursive Formulas for the Constants F2 (Fibonacci), F3 (Tribonacci), F4 (Tetranacci), and Other Constants of the Series.

This article proposes alternative recursive formulas for obtaining the constants F2 (Fibonacci), F3 (Tribonacci), F4 (Tetranacci), and other constants of this series, up to F11. Applying the formulas yields sequences in which dividing each number by the previous one leads in the limit to the required constants. Both ordinary formulas and sparse formulas "with a step" are considered, as well as formulas yielding the negative constants −F2…−F11. It is shown that in a number of cases the limit of the ratio yields not one but several alternating constants, whose product equals the corresponding constant of the series. The formulas are deliberately presented in tabular form so that they can be easily transferred to Excel and their results verified; known approaches to these constants rely on characteristic polynomials and Binet-type formulas, whereas the tabular sparse approach proposed here differs from them in form and is intended for direct verification in Excel. There are undoubtedly other alternative formulas as well, but the authors have restricted themselves to those presented in this work; the range of constants F2–F11 is limited by the page dimensions, since the tables cannot be extended in width.