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 A322261 #9 Dec 04 2018 15:26:48 %S A322261 0,1,1,2,2,2,3,5,5,3,4,6,10,4,4,5,9,13,11,11,5,6,10,18,12,20,10,6,7, %T A322261 13,21,19,27,21,9,7,8,14,26,20,36,26,22,8,8,9,17,29,27,43,37,25,23,23, %U A322261 9,10,18,34,28,52,42,38,24,40,22,10,11,21,37,35,59,53 %N A322261 Square array T(n, k) (n >= 0, k >= 0) read by antidiagonals upwards: the lengths of runs in binary expansion of T(n, k) correspond to the lengths of runs in binary expansion of n followed by the lengths of runs in binary expansion of k. %C A322261 The array T is associative. %H A322261 <a href="http://oeis.org/index/Bi#binary">Index entries for sequences related to binary expansion of n</a> %F A322261 T(n, 0) = T(0, n) = n. %F A322261 T(n, 1) = A042963(n+1). %F A322261 T(n, 2) = A047617(n+1). %F A322261 T(n, 3) = A047457(n+1). %F A322261 T(1, n) = A010078(n+1). %F A322261 T(2, n) = A004757(n) for any n > 0. %F A322261 A005811(T(n, k)) = A005811(n) + A005811(k). %F A322261 T(2*n, k) = A163621(2*n, k) for any n > 0 and k > 0. %F A322261 T(2*n, 2*n) = A020330(2*n) for any n > 0. %e A322261 Array T(n, k) begins (in decimal): %e A322261 n\k| 0 1 2 3 4 5 6 7 8 9 10 11 12 %e A322261 ---+-------------------------------------------------------- %e A322261 0| 0 1 2 3 4 5 6 7 8 9 10 11 12 %e A322261 1| 1 2 5 4 11 10 9 8 23 22 21 20 19 %e A322261 2| 2 5 10 11 20 21 22 23 40 41 42 43 44 %e A322261 3| 3 6 13 12 27 26 25 24 55 54 53 52 51 %e A322261 4| 4 9 18 19 36 37 38 39 72 73 74 75 76 %e A322261 5| 5 10 21 20 43 42 41 40 87 86 85 84 83 %e A322261 6| 6 13 26 27 52 53 54 55 104 105 106 107 108 %e A322261 7| 7 14 29 28 59 58 57 56 119 118 117 116 115 %e A322261 8| 8 17 34 35 68 69 70 71 136 137 138 139 140 %e A322261 Array T(n, k) begins (in binary): %e A322261 n\k | 0 1 10 11 100 101 110 111 1000 %e A322261 ----+--------------------------------------------------------------------------- %e A322261 0| 0 1 10 11 100 101 110 111 1000 %e A322261 1| 1 10 101 100 1011 1010 1001 1000 10111 %e A322261 10| 10 101 1010 1011 10100 10101 10110 10111 101000 %e A322261 11| 11 110 1101 1100 11011 11010 11001 11000 110111 %e A322261 100| 100 1001 10010 10011 100100 100101 100110 100111 1001000 %e A322261 101| 101 1010 10101 10100 101011 101010 101001 101000 1010111 %e A322261 110| 110 1101 11010 11011 110100 110101 110110 110111 1101000 %e A322261 111| 111 1110 11101 11100 111011 111010 111001 111000 1110111 %e A322261 1000| 1000 10001 100010 100011 1000100 1000101 1000110 1000111 10001000 %o A322261 (PARI) torl(n) = my (r=[]); while (n, r = concat(valuation(n+(n%2),2), r); n \= 2^r[1];); r %o A322261 fromrl(r) = my (v=0); for (i=1, #r, v = (v + (i%2))*2^r[i]-(i%2)); v %o A322261 T(n,k) = fromrl(concat(torl(n), torl(k))) %Y A322261 Cf. A004757, A005811, A010078, A020330, A042963, A047457, A047617, A101211, A163621. %K A322261 nonn,tabl,base %O A322261 0,4 %A A322261 _Rémy Sigrist_, Dec 01 2018