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 A372501 #38 May 12 2024 11:23:13 %S A372501 0,2,1,5,4,3,7,9,8,6,10,12,16,14,11,13,17,21,27,24,19,15,22,29,35,45, %T A372501 40,32,18,25,37,48,58,74,66,53,20,30,42,61,79,95,121,108,87,23,33,50, %U A372501 69,100,129,155,197,176,142,26,38,55,82,113,163,210,252,320,286,231 %N A372501 The 2-Zeckendorf array of the second kind, read by upward antidiagonals. %C A372501 The 2-Zeckendorf array of the second kind is based on the dual Zeckendorf representation of numbers (see A104326). %C A372501 Column k contains the numbers whose dual Zeckendorf expansion ends "... 0 1^(k-1)" where ^ denotes repetition. %C A372501 Rows satisfy this recurrence: T(n,k+1) = T(n,k) + T(n,k-1) + 2 for all n > 0 and k > 1. %C A372501 As a sequence, the array is a permutation of the nonnegative integers. %C A372501 As an array, T is an interspersion (hence also a dispersion). This holds as well for all Zeckendorf arrays of the second kind. %C A372501 In general, for the m-Zeckendorf array of the second kind, the row recursion is given by T(n,k) = T(n,k-1) + T(n,k-m) + m, and the first column represent the "even" numbers. %F A372501 T(n,1) = A090909(n+1). %F A372501 T(1,k) = A001911(k-1). %F A372501 T(2,k) = A019274(k-2). %F A372501 T(3,k) = A014739(k-1). %F A372501 T(n,1) = floor((n-1)*phi^2) and T(n,k+1) = floor((T(n,k)+1)*phi) for k > 0, where phi = (1+sqrt(5))/2. This can be considered as an alternative way to define the array. %e A372501 Array begins: %e A372501 k=1 2 3 4 5 6 7 %e A372501 +--------------------------------- %e A372501 n=1 | 0 1 3 6 11 19 32 %e A372501 n=2 | 2 4 8 14 24 40 66 %e A372501 n=3 | 5 9 16 27 45 74 121 %e A372501 n=4 | 7 12 21 35 58 95 155 %e A372501 n=5 | 10 17 29 48 79 129 210 %e A372501 n=6 | 13 22 37 61 100 163 265 %e A372501 n=7 | 15 25 42 69 113 184 299 %e A372501 The same in dual Zeckendorf form shows the pattern of digit suffixes, for example column k=3 is all numbers ending 011: %e A372501 k=1 2 3 4 %e A372501 +------------------------------ %e A372501 n=1 | 0 1 11 111 %e A372501 n=2 | 10 101 1011 10111 %e A372501 n=3 | 110 1101 11011 110111 %e A372501 n=4 | 1010 10101 101011 1010111 %e A372501 n=5 | 1110 11101 111011 1110111 %Y A372501 Cf. A104326. %Y A372501 Rows n=1..3: A001911, A019274, A014739. %Y A372501 Columns k=1..3: A090909, A276885, A188012. %Y A372501 Cf. k-th prepended column: A022342 (k=1), A023444 (k=2). %K A372501 nonn,tabl %O A372501 1,2 %A A372501 _A.H.M. Smeets_, May 03 2024