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 A075300 #51 Jun 01 2025 16:57:33
%S A075300 0,1,2,3,5,4,7,11,9,6,15,23,19,13,8,31,47,39,27,17,10,63,95,79,55,35,
%T A075300 21,12,127,191,159,111,71,43,25,14,255,383,319,223,143,87,51,29,16,
%U A075300 511,767,639,447,287,175,103,59,33,18,1023,1535,1279,895,575,351,207,119
%N A075300 Array A read by antidiagonals upwards: A(n, k) = array A054582(n,k) - 1 = 2^n*(2*k+1) - 1 with n,k >= 0.
%C A075300 From _Philippe Deléham_, Feb 19 2014: (Start)
%C A075300 A(0,k) = 2*k = A005843(k),
%C A075300 A(1,k) = 4*k + 1 = A016813(k),
%C A075300 A(2,k) = 8*k + 3 = A017101(k),
%C A075300 A(n,0) = A000225(n),
%C A075300 A(n,1) = A153893(n),
%C A075300 A(n,2) = A153894(n),
%C A075300 A(n,3) = A086224(n),
%C A075300 A(n,4) = A052996(n+2),
%C A075300 A(n,5) = A086225(n),
%C A075300 A(n,6) = A198274(n),
%C A075300 A(n,7) = A238087(n),
%C A075300 A(n,8) = A198275(n),
%C A075300 A(n,9) = A198276(n),
%C A075300 A(n,10) = A171389(n). (End)
%C A075300 A permutation of the nonnegative integers. - _Alzhekeyev Ascar M_, Jun 05 2016
%C A075300 The values in array row n, when expressed in binary, have n trailing 1-bits. - _Ruud H.G. van Tol_, Mar 18 2025
%H A075300 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A075300 From _Wolfdieter Lang_, Jan 31 2019: (Start)
%F A075300 Array A(n, k) = 2^n*(2*k+1) - 1, for n >= 0 and m >= 0.
%F A075300 The triangle is T(n, k) = A(n-k, k) = 2^(n-k)*(2*k+1) - 1, n >= 0, k=0..n.
%F A075300 See also A054582 after subtracting 1. (End)
%F A075300 From _Ruud H.G. van Tol_, Mar 17 2025: (Start)
%F A075300 A(0, k) is even. For n > 0, A(n, k) is odd and (3 * A(n, k) + 1) / 2 = A(n-1, 3*k+1).
%F A075300 A(n, k) = 2^n - 1 (mod 2^(n+1)) (equivalent to the comment about trailing 1-bits). (End)
%e A075300 The array A begins:
%e A075300 0 2 4 6 8 10 12 14 16 18 ...
%e A075300 1 5 9 13 17 21 25 29 33 37 ...
%e A075300 3 11 19 27 35 43 51 59 67 75 ...
%e A075300 7 23 39 55 71 87 103 119 135 151 ...
%e A075300 15 47 79 111 143 175 207 239 271 303 ...
%e A075300 31 95 159 223 287 351 415 479 543 607 ...
%e A075300 ... - _Philippe Deléham_, Feb 19 2014
%e A075300 From _Wolfdieter Lang_, Jan 31 2019: (Start)
%e A075300 The triangle T begins:
%e A075300 n\k 0 1 2 3 4 5 6 7 8 9 10 ...
%e A075300 0: 0
%e A075300 1: 1 2
%e A075300 2: 3 5 4
%e A075300 3: 7 11 9 6
%e A075300 4: 15 23 19 13 8
%e A075300 5 31 47 39 27 17 10
%e A075300 6: 63 95 79 55 35 21 12
%e A075300 7: 127 191 159 111 71 43 25 14
%e A075300 8: 255 383 319 223 143 87 51 29 16
%e A075300 9: 511 767 639 447 287 175 103 59 33 18
%e A075300 10: 1023 1535 1279 895 575 351 207 119 67 37 20
%e A075300 ...
%e A075300 T(3, 1) = 2^2*(2*1+1) - 1 = 12 - 1 = 11. (End)
%p A075300 A075300bi := (x,y) -> (2^x * (2*y + 1))-1;
%p A075300 A075300 := n -> A075300bi(A025581(n), A002262(n));
%p A075300 A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2);
%p A075300 A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))),2) - (n+1);
%t A075300 Table[(2^# (2 k + 1)) - 1 &[m - k], {m, 0, 10}, {k, 0, m}] (* _Michael De Vlieger_, Jun 05 2016 *)
%Y A075300 Inverse permutation: A075301. Transpose: A075302. The X-projection is given by A007814(n+1) and the Y-projection A025480.
%Y A075300 Cf. A002262, A025581, A054582, A241957.
%K A075300 nonn,tabl,easy
%O A075300 0,3
%A A075300 _Antti Karttunen_, Sep 12 2002