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 A367832 #55 Dec 11 2023 21:38:39 %S A367832 1,4,2,7,6,3,11,9,10,5,14,12,15,17,8,18,16,20,25,29,13,21,19,27,34,42, %T A367832 49,22,24,23,32,46,58,71,83,37,28,26,39,54,78,99,121,141,63,31,30,44, %U A367832 66,92,133,169,206,240,107,35,33,51,75,112,157,227,288,351,409,182,38,36,56,87,128,191,268 %N A367832 Array T(n, k) read by ascending antidiagonals is a dispersion based on A367467. Column 1 lists the numbers which cannot be represented by A367467(m) + m. For k >= 1, T(n, k+1) = A367467(T(n, k)) + T(n, k). %C A367832 This sequence is a permutation of the positive integers. %C A367832 The array T(n, k+1) - T(n, k) for k > 1 is also a permutation of the positive integers. %C A367832 Columns k > 2 together consist of all the numbers from A003152. These are all the positive numbers of the form floor(m*(1+1/sqrt(2))). %C A367832 In column 2 are all the numbers from A184119. These are all the numbers of the form floor((2+sqrt(2))*m - sqrt(2)/2). %C A367832 Column 2 together with the columns k > 2 are all the numbers from A087057; these are all the numbers of the form ceiling(m*sqrt(2)). Together with column 1, which consists of all the numbers from A083051, they cover all positive integers. %C A367832 An alternative definition that allows this array to be obtained without using A367467: %C A367832 Take for T(n, 1) and T(n, 2) the first and the second number which do not appear in any row r < n. Complete all rows by the recurrence T(n, k) = floor(T(n, k-1)*(1 + 1/sqrt(2))). Start in the first row with T(1, 1) = 1 and T(1, 2) = 2. %C A367832 Let Q(n, k) = T(n, k+2) - T(n, k+1) for k > 0. Let b(m) be the row n where the integer m is found in Q. Then we will obtain for (b(n)) the sequence: 1, 1, 1, 2, 1, 3, 2, 4, 1, 3, 5, 2, 6, 4, 1, ... . If we were to remove the first occurrence of each number in this sequence, we would get the same sequence again, hence (b(n)) is a fractal sequence. %D A367832 Clark Kimberling, Fractal sequences and interspersions, Ars Combinatoria 45 (1997) 157-168. %H A367832 Clark Kimberling, <a href="http://dx.doi.org/10.1090/S0002-9939-1993-1111434-0">Interspersions and dispersions</a>, Proceedings of the American Mathematical Society, 117 (1993) 313-321. %H A367832 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a> %F A367832 T(1, k) = A293078(k). %F A367832 T(n, 1) = A083051(n-1). %F A367832 T(n, 2) = A184119(n). %F A367832 Conjectured: T(n, 3) = A328987(n-1). %F A367832 T(1, k) = 2*T(1, k-1) - T(1, k-2) + floor(T(1, k-2)/2), for k > 2. %F A367832 T(n, k+1) = floor(T(n, k)*(1+1/sqrt(2))) for k > 1. %F A367832 T(n, k+1) = A367467(T(n, k)) + T(n, k). %e A367832 Array T(n, k) begins: %e A367832 1, 2, 3, 5, 8, 13, 22, 37, 63, 107, ... %e A367832 4, 6, 10, 17, 29, 49, 83, 141, 240, 409, ... %e A367832 7, 9, 15, 25, 42, 71, 121, 206, 351, 599, ... %e A367832 11, 12, 20, 34, 56, 99, 169, 288, 491, 839, ... %e A367832 14, 16, 27, 46, 78, 133, 227, 387, 660, 1126, ... %e A367832 18, 19, 32, 54, 92, 157, 268, 457, 780, 1331, ... %e A367832 21, 23, 39, 66, 112, 191, 326, 556, 949, 1620, ... %e A367832 ... %Y A367832 Cf. A083051, A087057, A184119, A293078, A328987, A367467. %Y A367832 Cf. A083050 (a closely related dispersion). %K A367832 nonn,tabl %O A367832 1,2 %A A367832 _Thomas Scheuerle_, Dec 02 2023 %E A367832 Edited by _Peter Munn_, Dec 11 2023