cp's OEIS Frontend

a(n) is well-defined since A356874(1) = 1, and A356874(n) < n for n >= 2.

a(n) is the smallest number k such that A364800(k) = n.

Number of partitions into distinct Fibonacci parts (1 counted as two distinct Fibonacci numbers).

Inverse Euler transform of sequence has generating function sum_{n>0} x^F(n)-x^{2F(n)} where F() is Fibonacci.

From Peter Munn, Apr 28 2025: (Start)

Each row in the Wythoff array, A035513, and this extended array satisfies the Fibonacci recurrence; that is each term after the first 2 is the sum of the preceding 2 terms.

We use F_i to denote the i-th Fibonacci term, A000045(i). In particular, we refer below to F_0 = 0, F_1 = 1 and F_2 = 1 several times. Note that to fully understand the description of the relationship between neighboring columns it is important to distinguish F_1 and F_2, although they have the same integer value. Similarly, the identity of an array term should be understood here as including its position in the array, not only its integer value.

The terms of this extended Wythoff array map 1:1 onto the nonempty finite subsets of Fibonacci terms (from F_0 onwards) that do not include both F_i and F_{i+1} for any i. With this map each term is the sum of its subset image. See the table in the examples.

Full description of the mapping with its relationship to A035513:

The (unextended) Wythoff array A035513 includes every positive integer exactly once. So, using the Zeckendorf representation (see link below), the array terms map 1:1 to nonempty finite subsets of the Fibonacci terms from F_2 onwards -- more precisely, onto those that do not include both F_i and F_{i+1} for any i. (Again, each array term is the sum of the Fibonacci numbers from the relevant subset.)

As shown in the Kimberling 1995 link, when we proceed from one term to the next in a row, the indices of the Fibonacci terms in the corresponding subset are incremented. When we proceed leftwards, the indices are decremented, with the subsets for the leftmost column being those that include F_2.

And when we add 2 columns on the left of the Wythoff array, the mapping continues to decrement the indices, so the corresponding extra subsets have F_0 (new leftmost column) or F_1 as their first Fibonacci term.

Thus the terms of this extended Wythoff array map 1:1 onto the nonempty finite subsets of Fibonacci terms (from F_0 onwards) that do not include both F_i and F_{i+1} for any i. The leftmost column is the nonnegative integers: if we were to remove F_0 (value 0) from the subset for an integer in this column, the subset would form the Zeckendorf representation of the integer, as subsets do in the unextended array.

(End)

The nonzero Fibbinary numbers (A003714) arranged in rows where each successive term is twice the preceding term; a (transposed) Fibbinary equivalent of A054582.

Write the first term in each row as Sum_{i in S} 2^i, where S is a set of nonnegative integers, then n = Sum_{i in S} F_i, where F_i is the i-th Fibonacci number, A000045(i).

More generally, if the terms are represented in binary, and the binary weighting of the digits (2^0, 2^1, 2^2, ...) is replaced with Fibonacci weighting (F_0, F_1, F_2, ...), we get the extended Wythoff array (A287870). If the weighting of the Zeckendorf representation is used (F_2, F_3, F_4, ...), we get the (unextended) Wythoff array (A035513).

Note the Zeckendorf representation of 0 is taken to be the empty sum.

The Wythoff array A035513 is the subtable formed by rows 3, 11, 16, 24, 32, ... (A035337). If, instead, we use rows 2, 7, 10, 15, 20, ... (A035336) or 1, 4, 6, 9, 12, ... (A003622), we get the Wythoff array extended by 1 column (A287869) or 2 columns (A287870) respectively.

Similarly, using A035338 truncates by 1 column; and in general if S_k is column k of the Wythoff array then the rows here numbered by S_k form an array A_k that starts with column k-2 of the Wythoff array. (A_0 and A_1 are the 2 extended arrays mentioned above.) As every positive integer occurs exactly once in the Wythoff array, every row except row 0 of A(.,.) is a row of exactly one such A_k.

Columns 4 onwards match certain columns of the multiplication table for Knuth's Fibonacci (or circle) product (extended variant - see A135090 and formula below).

For k > 0, the first row to contain k is A348853(k).