By Robert C. Johnson
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Obvious generalisation: to ‘tribonacci polynomials’ with 2 x x 1 ξ2 1 0 0 and , 1 − x2 ξ − xξ 2 − ξ 3 0 1 0 . . and so on. 46 Appendix B Formulas There are collections of standard Fibonacci and Lucas formulas online at MathWorld [3] and at Fibonacci Numbers, the Golden Section and the Golden String [33], denoted respectively as ‘Eric’ and ‘Ron’. Eric’s collection is the more sophisticated. Any candidate relation can be checked either by induction or with the Binet formulas1 eq. 8) and eq.
4) as follows — • Lucas’ duplication formula: (2, 2n, n + 1, n + 1, 2), • Vadja-11, Dunlap-7: (n + 1, −n − 1, n, −n, n), • Vadja-20a: (n + i, n + k, n, n + i + k, n), and it’s pleasant exercise to deal with the rest likewise . . however . . • Vadja-12 (and its analogue with f → ) comes most naturally direct from eq. 1) — multiply fn−1 = fn+1 − fn by fn+2 = fn+1 + fn . • Dunlap-10 is just eq. 2) with (k, l) → (m, n − m). To illustrate the Lucas formulas, take (a, b, c, d, r) in the eq. 23) as follows — remembering that ( 0 , 1 ) = (2, 1) — version of • ‘Cassini’ (see eq.
23) as follows — remembering that ( 0 , 1 ) = (2, 1) — version of • ‘Cassini’ (see eq. 8)): (n + 1, n − 1, n, n, n + 1), • Vadja-17a: (n + m, 0, n, m, m), • Vadja-17c (duplication): (2n, 0, n, n, n). Among the Fibonacci and Lucas results — • Vadja-13 comes from eq. 4) with (a, b, c, d, r) → (2n, 1, n, n + 1, n) , using (f0 , f1 ) = (0, 1) plus eq. 1) and eq. 1). See also Sec. 3. • Vadja-15a follows from the f version of eq. 23) with (a, b, c, d, r) → (n, m, 0, m + n, m) , plus eq. 2). Otherwise from the mixed formula eq.