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 A054582 #69 Feb 04 2019 11:22:10
%S A054582 1,2,3,4,6,5,8,12,10,7,16,24,20,14,9,32,48,40,28,18,11,64,96,80,56,36,
%T A054582 22,13,128,192,160,112,72,44,26,15,256,384,320,224,144,88,52,30,17,
%U A054582 512,768,640,448,288,176,104,60,34,19,1024,1536,1280,896,576,352,208,120
%N A054582 Array read by antidiagonals upwards: A(m,k) = 2^m * (2k+1), m,k >= 0.
%C A054582 First column of array is powers of 2, first row is odd numbers, other cells are products of these two, so every positive integer appears exactly once. [Comment edited to match the definition. - _L. Edson Jeffery_, Jun 05 2015]
%C A054582 An analogous N X N <-> N bijection based, not on the binary, but on the Fibonacci number system, is given by the Wythoff array A035513.
%C A054582 As an array, this sequence (hence also A135764) is the dispersion of the even positive integers. For the definition of dispersion, see the link "Interspersions and Dispersions." The fractal sequence of this dispersion is A003602. - _Clark Kimberling_, Dec 03 2010
%H A054582 Reinhard Zumkeller, <a href="/A054582/b054582.txt">Rows n = 0..100 of triangle, flattened</a>
%H A054582 Clark Kimberling, <a href="http://faculty.evansville.edu/ck6/integer/intersp.html">Interspersions and Dispersions</a>.
%H A054582 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A054582 As a sequence, if n is a triangular number, then a(n)=a(n-A002024(n))+2, otherwise a(n)=2*a(n-A002024(n)-1).
%F A054582 a(n) = A075300(n-1)+1.
%F A054582 Recurrence for the sequence: if a(n-1)=2*k is even, then a(n)=k+A006519(2*k); if a(n-1)=2*k+1 is odd, then a(n)=2^(k+1), a(0)=1. - _Philippe Deléham_, Dec 13 2013
%F A054582 m = A(A001511(m)-1, A003602(m)-1), for each m in A000027. - _L. Edson Jeffery_, Nov 22 2015
%F A054582 The triangle is T(n, k) = A(n-k, k) = 2^(n-k)*(2*k+1), for n >= 0 and k = 0..n. - _Wolfdieter Lang_, Jan 30 2019
%e A054582 Northwest corner of array A:
%e A054582 1 3 5 7 9 11 13 15 17 19
%e A054582 2 6 10 14 18 22 26 30 34 38
%e A054582 4 12 20 28 36 44 52 60 68 76
%e A054582 8 24 40 56 72 88 104 120 136 152
%e A054582 16 48 80 112 144 176 208 240 272 304
%e A054582 32 96 160 224 288 352 416 480 544 608
%e A054582 64 192 320 448 576 704 832 960 1088 1216
%e A054582 128 384 640 896 1152 1408 1664 1920 2176 2432
%e A054582 256 768 1280 1792 2304 2816 3328 3840 4352 4864
%e A054582 512 1536 2560 3584 4608 5632 6656 7680 8704 9728
%e A054582 [Array edited to match the definition. - _L. Edson Jeffery_, Jun 05 2015]
%e A054582 From _Philippe Deléham_, Dec 13 2013: (Start)
%e A054582 a(13-1)=20=2*10, so a(13)=10+A006519(20)=10+4=14.
%e A054582 a(3-1)=3=2*1+1, so a(3)=2^(1+1)=4. (End)
%e A054582 From _Wolfdieter Lang_, Jan 30 2019: (Start)
%e A054582 The triangle T begins:
%e A054582 n\k 0 1 2 3 4 5 6 7 8 9 10 ...
%e A054582 0: 1
%e A054582 1: 2 3
%e A054582 2: 4 6 5
%e A054582 3: 8 12 10 7
%e A054582 4: 16 24 20 14 9
%e A054582 5: 32 48 40 28 18 11
%e A054582 6: 64 96 80 56 36 22 13
%e A054582 7: 128 192 160 112 72 44 26 15
%e A054582 8: 256 384 320 224 144 88 52 30 17
%e A054582 9: 512 768 640 448 288 176 104 60 34 19
%e A054582 10: 1024 1536 1280 896 576 352 208 120 68 38 21
%e A054582 ...
%e A054582 T(3, 2) = 2^1*(2*2+1) = 10. (End)
%t A054582 (* Array: *)
%t A054582 Grid[Table[2^m*(2*k + 1), {m, 0, 9}, {k, 0, 9}]] (* _L. Edson Jeffery_, Jun 05 2015 *)
%t A054582 (* Array antidiagonals flattened: *)
%t A054582 Flatten[Table[2^(m - k)*(2*k + 1), {m, 0, 9}, {k, 0, m}]] (* _L. Edson Jeffery_, Jun 05 2015 *)
%o A054582 (Haskell)
%o A054582 a054582 n k = a054582_tabl !! n !! k
%o A054582 a054582_row n = a054582_tabl !! n
%o A054582 a054582_tabl = iterate
%o A054582 (\xs@(x:_) -> (2 * x) : zipWith (+) xs (iterate (`div` 2) (2 * x))) [1]
%o A054582 a054582_list = concat a054582_tabl
%o A054582 -- _Reinhard Zumkeller_, Jan 22 2013
%o A054582 (PARI) T(m,k)=(2*k+1)<<m \\ _Charles R Greathouse IV_, Jun 21 2017
%Y A054582 The sequence is a permutation of A000027.
%Y A054582 Main diagonal is A014480; inverse permutation is A209268.
%Y A054582 Cf. A001511, A002024, A003602, A006519, A025480, A035513, A075300, A135764.
%K A054582 easy,nice,nonn,tabl
%O A054582 0,2
%A A054582 _Henry Bottomley_, Apr 12 2000
%E A054582 Offset corrected by _Reinhard Zumkeller_, Jan 22 2013