This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A287870 #46 May 03 2025 09:42:14
%S A287870 0,1,1,1,3,2,2,4,4,3,3,7,6,6,4,5,11,10,9,8,5,8,18,16,15,12,9,6,13,29,
%T A287870 26,24,20,14,11,7,21,47,42,39,32,23,17,12,8,34,76,68,63,52,37,28,19,
%U A287870 14,9,55,123,110,102,84,60,45,31,22,16,10,89,199,178,165,136,97,73,50,36,25,17,11
%N A287870 The extended Wythoff array (the Wythoff array with two extra columns) read by antidiagonals downwards.
%C A287870 From _Peter Munn_, Apr 28 2025: (Start)
%C A287870 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.
%C A287870 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.
%C A287870 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.
%C A287870 Full description of the mapping with its relationship to A035513:
%C A287870 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.)
%C A287870 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.
%C A287870 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.
%C A287870 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.
%C A287870 (End)
%H A287870 Peter G. Anderson, <a href="https://www.fq.math.ca/Papers1/52-5/Anderson.pdf">More Properties of the Zeckendorf Array</a>, Fib. Quart. 52-5 (2014), 15-21.
%H A287870 John Conway and Alex Ryba, <a href="https://doi.org/10.1007/s00283-015-9582-5">The extra Fibonacci series and the Empire State Building</a>, Math. Intelligencer 38 (2016), no. 1, 41-48. See <a href="https://www.researchgate.net/publication/291951643_The_Extra_Fibonacci_Series_and_the_Empire_State_Building">preview</a>, at ResearchGate.
%H A287870 Encyclopedia of Mathematics, <a href="https://encyclopediaofmath.org/wiki/Zeckendorf_representation">Zeckendorf representation</a>
%H A287870 Clark Kimberling, <a href="http://www.fq.math.ca/Scanned/33-1/kimberling.pdf">The Zeckendorf array equals the Wythoff array</a>, Fibonacci Quarterly, Vol. 33, No. 1 (1995), pp. 3-8.
%F A287870 From _Peter Munn_, Apr 29 2025: (Start)
%F A287870 A(n,k) = A356874(floor(m/2)), where m = A356875(n-1, k-1) = A054582(k-1, (A022341(n-1)-1)/2).
%F A287870 A(n,k) = A357316(A003622(n), k-1).
%F A287870 (End)
%e A287870 The extended Wythoff array is the Wythoff array with two extra columns, giving the row number n and A000201(n), separated from the main array by a vertical bar:
%e A287870 0 1 | 1 2 3 5 8 13 21 34 55 89 144 ...
%e A287870 1 3 | 4 7 11 18 29 47 76 123 199 322 521 ...
%e A287870 2 4 | 6 10 16 26 42 68 110 178 288 466 754 ...
%e A287870 3 6 | 9 15 24 39 63 102 165 267 432 699 1131 ...
%e A287870 4 8 | 12 20 32 52 84 136 220 356 576 932 1508 ...
%e A287870 5 9 | 14 23 37 60 97 157 254 411 665 1076 1741 ...
%e A287870 6 11 | 17 28 45 73 118 191 309 500 809 1309 2118 ...
%e A287870 7 12 | 19 31 50 81 131 212 343 555 898 1453 2351 ...
%e A287870 8 14 | 22 36 58 94 152 246 398 644 1042 1686 2728 ...
%e A287870 9 16 | 25 41 66 107 173 280 453 733 1186 1919 3105 ...
%e A287870 10 17 | 27 44 71 115 186 301 487 788 1275 2063 3338 ...
%e A287870 11 19 | 30 49 79 ...
%e A287870 12 21 | 33 54 87 ...
%e A287870 13 22 | 35 57 92 ...
%e A287870 14 24 | 38 62 ...
%e A287870 15 25 | 40 65 ...
%e A287870 16 27 | 43 70 ...
%e A287870 17 29 | 46 75 ...
%e A287870 18 30 | 48 78 ...
%e A287870 19 32 | 51 83 ...
%e A287870 20 33 | 53 86 ...
%e A287870 21 35 | 56 91 ...
%e A287870 22 37 | 59 96 ...
%e A287870 23 38 | 61 99 ...
%e A287870 24 40 | 64 ...
%e A287870 25 42 | 67 ...
%e A287870 26 43 | 69 ...
%e A287870 27 45 | 72 ...
%e A287870 28 46 | 74 ...
%e A287870 29 48 | 77 ...
%e A287870 30 50 | 80 ...
%e A287870 31 51 | 82 ...
%e A287870 32 53 | 85 ...
%e A287870 33 55 | 88 ...
%e A287870 34 56 | 90 ...
%e A287870 35 58 | 93 ...
%e A287870 36 59 | 95 ...
%e A287870 37 61 | 98 ...
%e A287870 38 63 | ...
%e A287870 ...
%e A287870 From _Peter Munn_, Sep 12 2022: (Start)
%e A287870 In the table below, the array terms are shown in the small box at the bottom right of the cells. At the top of each cell is shown a pattern of Fibonacci terms, with "*" indicating a Fibonacci term that appears below it. Those Fibonacci terms sum to the array term. The pattern never includes "**", which would indicate 2 consecutive Fibonacci terms. Note that a Fibonacci term shown as "1" in the 2nd column is F_1, so it may accompany "2", which is F_3. In other columns a Fibonacci term shown as "1" is F_2 and may not accompany "2".
%e A287870 +----------+-----------+------------+------------+------------+
%e A287870 | * | * | * | * | * |
%e A287870 | 0 __| 1 ___| 1 ___| 2 ___| 3 ___|
%e A287870 | |0 | | 1 | | 1 | | 2 | | 3 |
%e A287870 |----------+-----------+------------+------------+------------|
%e A287870 | * * | * * | * * | * * | * * |
%e A287870 | 0 __| 1 ___| 1 ___| 2 ___| 3 ___|
%e A287870 | 1 |1 | 2 | 3 | 3 | 4 | 5 | 7 | 8 |11 |
%e A287870 |----------+-----------+------------+------------+------------|
%e A287870 | * * | * * | * * | * * | * * |
%e A287870 | 2 0 __| 3 1 ___| 5 1 ___| 8 2 ___| 13 3 ___|
%e A287870 | |2 | | 4 | | 6 | |10 | |16 |
%e A287870 |----------+-----------+------------+------------+------------|
%e A287870 | * * | * * | * * | * * | * * |
%e A287870 | 0 __| 1 ___| 1 ___| 2 ___| 3 ___|
%e A287870 | 3 |3 | 5 | 6 | 8 | 9 | 13 |15 | 21 |24 |
%e A287870 |----------+-----------+------------+------------+------------|
%e A287870 | * * * | * * * | * * * | * * * | * * * |
%e A287870 | 0 | 1 | 1 | 2 | 3 |
%e A287870 | 1 __| 2 ___| 3 ___| 5 ___| 8 ___|
%e A287870 | 3 |4 | 5 | 8 | 8 |12 | 13 |20 | 21 |32 |
%e A287870 |----------+-----------+------------+------------+------------|
%e A287870 | * * | * * | * * | * * | * * |
%e A287870 | 0 __| 1 ___| 1 ___| 2 ___| 3 ___|
%e A287870 | 5 |5 | 8 | 9 | 13 |14 | 21 |23 | 34 |37 |
%e A287870 |----------+-----------+------------+------------+------------|
%e A287870 | * * * | * * * | * * * | * * * | * * * |
%e A287870 | 0 __| 1 ___| 1 ___| 2 ___| 3 ___|
%e A287870 | 5 1 |6 | 8 2 |11 | 13 3 |17 | 21 5 |28 | 34 8 |45 |
%e A287870 |----------+-----------+------------+------------+------------|
%e A287870 | * * * | * * * | * * * | * * * | * * * |
%e A287870 | 2 0 __| 3 1 ___| 5 1 ___| 8 2 ___| 13 3 ___|
%e A287870 | 5 |7 | 8 |12 | 13 |19 | 21 |31 | 34 |50 |
%e A287870 +----------+-----------+------------+------------+------------+
%e A287870 If we replace the Fibonacci terms 0, 1, 1, 2, 3, 5, ... in the main part of the cells with the powers of 2 (1, 2, 4, ...) the sums in the small boxes become the terms of A356875. From this may be seen a relationship to A054582.
%e A287870 - - - - -
%e A287870 Each row of the extended Wythoff array satisfies the Fibonacci recurrence, and may be further extended to the left using this recurrence backwards:
%e A287870 ... -1 1 0 1 | 1 2 3 5 ...
%e A287870 ... -1 2 1 3 | 4 7 11 18 ...
%e A287870 ... 0 2 2 4 | 6 10 16 26 ...
%e A287870 ... 0 3 3 6 | 9 15 24 39 ...
%e A287870 ... 0 4 4 8 | 12 20 32 52 ...
%e A287870 ... 1 4 5 9 | 14 23 37 60 ...
%e A287870 ... 1 5 6 11 | 17 28 45 73 ...
%e A287870 ... 2 5 7 12 | 19 31 50 81 ...
%e A287870 ... 2 6 8 14 | 22 36 58 94 ...
%e A287870 ...
%e A287870 ... 5 10 15 25 | 40 65 105 170 ...
%e A287870 ...
%e A287870 Note that multiples (*2, *3 and *4) of the top (Fibonacci sequence) row appear a little below, but shifted 2 columns to the left. Larger multiples appear further down and shifted further to the left, starting with row 15, where the terms are 5 times those in the top row and shifted 4 columns leftwards.
%e A287870 (End)
%Y A287870 Subtables: A035513 (the Wythoff array), A287869.
%Y A287870 Related as a subtable of A357316 as A054582 is to A130128 (as a square).
%Y A287870 Cf. A000045, A000201.
%Y A287870 See A014417 for sequences related to Zeckendorf representation.
%Y A287870 See the formula section for the relationships with A003622, A022341, A054582, A356874, A356875.
%K A287870 nonn,tabl
%O A287870 1,5
%A A287870 _N. J. A. Sloane_, Jun 14 2017