One theorem about these numbers states that every natural number can be represented uniquely as the sum of nonconsecutive ones of them; that statement is named for Zeckendorf. This sequence appears on the outer edges of Hosoya's triangle, and its nth element gives the number
One theorem about these numbers states that every natural number can be represented uniquely as the sum of nonconsecutive ones of them; that statement is named for Zeckendorf. This sequence appears on the outer edges of Hosoya's triangle, and its nth element gives the number of ways for dominoes to cover a two-by-(n minus one) checkerboard. This sequence is generated by a formula named for (*) Binet, as well as by the function x over the quantity one minus x minus x-squared. It is a specific type of Lucas sequence, and the ratio of consecutive elements of this sequence approaches the golden ratio. First proposed in Liber Abaci, for 10 points, name this recursively-defined sequence whose elements are the sum of the previous two elements, beginning 0, 1, 1, 2, 3, 5, 8.
One theorem about these numbers states that every natural number can be represented uniquely as the sum of nonconsecutive ones of them; that statement is named for Zeckendorf.
{Fibonacci} sequence/numbers/series [prompt on {Lucas sequence} before it is mentioned; prompt on {pine cone} sequence/numbers/series; do not prompt on “{Lucas} numbers”]