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 A357316 #11 Aug 18 2025 16:16:37
%S A357316 0,0,0,0,1,1,0,1,1,1,0,2,2,2,1,0,3,3,3,3,2,0,5,5,5,4,3,2,0,8,8,8,7,5,
%T A357316 4,3,0,13,13,13,11,8,6,4,3,0,21,21,21,18,13,10,7,5,3,0,34,34,34,29,21,
%U A357316 16,11,8,6,4,0,55,55,55,47,34,26,18,13,9,6,4
%N A357316 A distension of the Wythoff array by inclusion of intermediate rows. Square array A(n,k), n >= 0, k >= 0, read by descending antidiagonals. If S is the set such that Sum_{i in S} F_i is the Zeckendorf representation of n then A(n,k) = Sum_{i in S} F_{i+k-2}.
%C A357316 Note the Zeckendorf representation of 0 is taken to be the empty sum.
%C A357316 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.
%C A357316 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.
%C A357316 Columns 4 onwards match certain columns of the multiplication table for Knuth's Fibonacci (or circle) product (extended variant - see A135090 and formula below).
%C A357316 For k > 0, the first row to contain k is A348853(k).
%H A357316 Encyclopedia of Mathematics, <a href="https://encyclopediaofmath.org/wiki/Zeckendorf_representation">Zeckendorf representation</a>
%F A357316 For n >= 0, k >= 0 unless stated otherwise:
%F A357316 A(n,k) = A356874(floor(A003714(n)*2^(k-1))).
%F A357316 A(n,1) = A005206(n).
%F A357316 A(n,2) = n.
%F A357316 A(n,k+2) = A(n,k) + A(n,k+1).
%F A357316 A(A022342(n+1),k) = A(n,k+1).
%F A357316 For k >= 4, A(n,k) = A135090(n,A000045(k-2)).
%e A357316 Example for n = 4, k = 3. The Zeckendorf representation of 4 is F_4 + F_2 = 3 + 1. So the values of i in the sums in the definition are 4 and 2; hence A(4,3) = Sum_{i = 2,4} F_{i+k-2} = F_{4+3-2} + F_{2+3-2} = F_5 + F_3 = 5 + 2 = 7.
%e A357316 Square array A(n,k) begins:
%e A357316 n\k| 0 1 2 3 4 5 6
%e A357316 ----+--------------------------------
%e A357316 0 | 0 0 0 0 0 0 0 ...
%e A357316 1* | 0 1 1 2 3 5 8 ...
%e A357316 2 | 1 1 2 3 5 8 13 ...
%e A357316 3 | 1 2 3 5 8 13 21 ...
%e A357316 4* | 1 3 4 7 11 18 29 ...
%e A357316 5 | 2 3 5 8 13 21 34 ...
%e A357316 6* | 2 4 6 10 16 26 42 ...
%e A357316 7 | 3 4 7 11 18 29 47 ...
%e A357316 8 | 3 5 8 13 21 34 55 ...
%e A357316 9* | 3 6 9 15 24 39 63 ...
%e A357316 10 | 4 6 10 16 26 42 68 ...
%e A357316 11 | 4 7 11 18 29 47 76 ...
%e A357316 12* | 4 8 12 20 32 52 84 ...
%e A357316 ...
%e A357316 The asterisked rows form the start of the extended Wythoff array (A287870).
%o A357316 (PARI)
%o A357316 A5206(m) = if(m>0,m-A5206(A5206(m-1)),0)
%o A357316 A(n,k) = if(k==2,n, if(k==1,A5206(n), if(k==0,n-A5206(n), A(n,k-2)+A(n,k-1)))) \\ simple encoding of formulas, not efficient
%Y A357316 Columns, some differing initially: A005206 (1), A022342 (3), A026274 (4), A101345 (5), A101642 (6).
%Y A357316 Rows: A000045 (1), A000204 (4).
%Y A357316 Related to subtable A287870 as A130128 (as a square) is to A054582.
%Y A357316 Other subtables: A035513, A287869.
%Y A357316 See the comments for the relationship to A003622, A035336, A035337, A035338, A348853.
%Y A357316 See the formula section for the relationship to A003714, A022342, A135090, A356874.
%K A357316 nonn,tabl
%O A357316 0,12
%A A357316 _Peter Munn_, Sep 23 2022