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 A136189 #28 Apr 08 2025 23:24:55 %S A136189 1,2,5,3,8,7,4,12,11,10,6,17,16,15,14,9,25,23,22,21,18,13,37,34,32,31, %T A136189 27,20,19,54,50,47,45,40,30,24,28,79,73,69,66,58,44,36,26,41,116,107, %U A136189 101,97,85,64,53,39,29,60,170,157,148,142,125,94,77,57,43,33,88,249,230 %N A136189 The 3rd-order Zeckendorf array, T(n,k), read by antidiagonals. %C A136189 Rows satisfy this recurrence: T(n,k) = T(n,k-1) + T(n,k-3) for all k>=4. %C A136189 Except for initial terms, (row 1) = A000930 (column 1) = A020942 (column 2) = A064105 (column 3) = A064106. %C A136189 As a sequence, the array is a permutation of the natural numbers. %C A136189 As an array, T is an interspersion (hence also a dispersion). %H A136189 Clark Kimberling, <a href="https://www.fq.math.ca/Scanned/33-1/kimberling.pdf">The Zeckendorf array equals the Wythoff array</a>, Fibonacci Quarterly 33 (1995) 3-8. %H A136189 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a> %F A136189 Row 1 is the 3rd-order Zeckendorf basis, given by initial terms b(1)=1, b(2)=2, b(3)=3 and recurrence b(k) = b(k-1) + b(k-3) for k>=4. Every positive integer has a unique 3-Zeckendorf representation: n = b(i(1)) + b(i(2)) + ... + b(i(p)), where |i(h)-i(j)| >= 3. Rows of T are defined inductively: T(n,1) is the least positive integer not in an earlier row. T(n,2) is obtained from T(n,1) as follows: if T(n,1) = b(i(1)) + b(i(2)) + ... + b(i(p)), then T(n,k+1) = b(i(1+k)) + b(i(2+k)) + ... + b(i(p+k)) for k=1,2,3,... . %F A136189 A(n, k) = A000930(k)*A202342(n) + A000930(k-2)*A136495(n) + A000930(k-1)*(n-1) for n > 1. - _Alan Michael Gómez Calderón_, Dec 23 2024 %e A136189 Northwest corner: %e A136189 1 2 3 4 6 9 13 19 ... %e A136189 5 8 12 17 25 37 54 79 ... %e A136189 7 11 16 23 34 50 73 107 ... %e A136189 10 15 22 32 47 69 101 148 ... %e A136189 ... %Y A136189 Cf. A000930, A035513, A134563, A136175, A136495, A202342. %K A136189 nonn,tabl %O A136189 1,2 %A A136189 _Clark Kimberling_, Dec 20 2007