The Fibonacci Sequence, Fibonacci Assoc., Santa Clara, CA, 1980, pp. Morrison, A Stolarsky array of Wythoff pairs, in A Collection of Manuscripts Related to Kimberling, Fractal Sequences and Interspersions, Ars Combinatoria, vol 45 p 157 1997. Kimberling, The Zeckendorf array equals the Wythoff array, Fibonacci Quarterly 33 (1995) 3-8. Kimberling, Stolarsky interspersions, Ars Combinatoria 39 (1995) 129-138. Kimberling, Numeration systems and fractal sequences, Acta Arithmetica 73 (1995) 103-117. Kimberling, The First Column of an Interspersion, Fibonacci Quarterly 32 (1994) 301-314. Kimberling, Interspersions and dispersions, Proc. Bergum et al., editors, Applications of Fibonacci Numbers, Vol. Kimberling, Orderings of the set of all positive Fibonacci sequences, Kimberling, Problem 1615, Crux Mathematicorum, Vol. Kimberling, Generalized Wythoff arrays, shuffles and interspersions, This sequence also has a very nice property Gives the index (or parameter) of the column of the WythoffĪrray that contains n. Nest, so the sequence can be rewritten as: We see a permutation of the first few positive integers, and these If you delete the first occurrence of each number, the This sequence also has some nice properties.Ī. Gives the index (or parameter) of the row of the Wythoff array that contains n. Of the numbers whose Zeckendorf expansion ends The kth column of the Wythoff array consists. Īnother especially interesting array with properties 1-7 is the Stolarsky array: There are infinitely many arrays with properties 1-7, see. satisfyingĪ(n)=a(n-1)+a(n-2) and eventually positive) appears Every positive Fibonacci-type sequence (i.e. The terms in any row or column are monotonically increasing Ħ. Every positive integer appears exactly once in the array ĥ. The leading term in each row is the smallest number not foundĤ. Every row satisfies the Fibonacci recurrence ģ. The first row of the Wythoff array consists of theĢ. The broken line read n, 1+Sn then after the broken Construction (2): the two columns to the left of.The Fibonacci successor to (or left shift of) n,Įxpansion by F i+1 for example the successor to 100 is S100 = 144 + 13 + 5 = 162. Largest Fibonacci number you can until nothing remains for example 100 = 89 + 8 + 3 The Zeckendorf expansion of n is obtained by repeatedly subtracting the The entry n in the first column is the index That each term is the sum of the two previous terms. The rows are then filled in by the Fibonacci rule Whose nth term is, where tau=(1+sqrt(5))/2. The broken line consist respectively of the nonnegative integers n, and Construction (1): the two columns to the left of.(For sources see the "References" below.) To a large number of sequences in the On-Line Encyclopedia.Ģ 4 | 6 10 16 26 42 68 110 178 288 466 754 It has many wonderful properties, some of Is shown below, to the right of the broken The Wythoff Array and The Para-Fibonacci Sequence Classic Sequences In The On-Line Encyclopedia of Integer Sequences® (OEIS®)
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