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 A257852 #29 Apr 24 2024 13:58:29
%S A257852 3,1,7,13,9,11,5,29,17,15,53,37,45,25,19,21,117,69,61,33,23,213,149,
%T A257852 181,101,77,41,27,85,469,277,245,133,93,49,31,853,597,725,405,309,165,
%U A257852 109,57,35,341,1877,1109,981,533,373,197,125,65,39
%N A257852 Array A read by upward antidiagonals in which the entry A(n,k) in row n and column k is defined by A(n,k) = (2^n*(6*k - 3 - 2*(-1)^n) - 1)/3, n,k >= 1.
%C A257852 Sequence is a permutation of the odd natural numbers.
%C A257852 Let N_1 denote the set of odd natural numbers, and let |y|_2 denote 2-adic valuation of y. Define the map F : N_1 -> N_1 by F(x) = (3*x + 1)/2^|3*x+1|_2 (cf. A075677). Then row n of A is the set of all x in N_1 for which |3*x + 1|_2 = A371093(x) = n. Hence F(A(n,k)) = 6*k - 3 - 2*(-1)^n.
%H A257852 Antti Karttunen, <a href="/A257852/b257852.txt">Table of n, a(n) for n = 1..10440; the first 144 antidiagonals, flattened</a>
%H A257852 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%F A257852 From _Ruud H.G. van Tol_, Oct 17 2023: (Start)
%F A257852 A(n,k+1) = A(n,k) + 2^(n+1).
%F A257852 A(n+2,k) = A(n,k)*4 + 1.
%F A257852 A(1,k) = A004767(k-1).
%F A257852 A(2,k) = A017077(k-1).
%F A257852 A(3,k) = A082285(k-1).
%F A257852 A(4,k) = A238477(k). (End)
%F A257852 For all odd positive numbers n, A(A371093(n), A371092(n)) = n. - _Antti Karttunen_, Apr 24 2024
%e A257852 From _Ruud H.G. van Tol_, Oct 17 2023, corrected and extended by _Antti Karttunen_, Apr 18 2024: (Start)
%e A257852 Array A begins:
%e A257852 n\k| 1| 2| 3| 4| 5| 6| 7| 8| ...
%e A257852 ---+---------------------------------------------
%e A257852 1 | 3, 7, 11, 15, 19, 23, 27, 31, ...
%e A257852 2 | 1, 9, 17, 25, 33, 41, 49, 57, ...
%e A257852 3 | 13, 29, 45, 61, 77, 93, 109, 125, ...
%e A257852 4 | 5, 37, 69, 101, 133, 165, 197, 229, ...
%e A257852 5 | 53, 117, 181, 245, 309, 373, 437, 501, ...
%e A257852 6 | 21, 149, 277, 405, 533, 661, 789, 917, ...
%e A257852 ... (End)
%t A257852 (* Array: *)
%t A257852 Grid[Table[(2^n*(6*k - 3 - 2*(-1)^n) - 1)/3, {n, 10}, {k, 10}]]
%t A257852 (* Array antidiagonals flattened: *)
%t A257852 Flatten[Table[(2^(n - k + 1)*(6*k - 3 - 2*(-1)^(n - k + 1)) - 1)/ 3, {n, 10}, {k, n}]]
%o A257852 (PARI)
%o A257852 up_to = 105;
%o A257852 A257852sq(n,k) = ((2^n * (6*k - 3 - 2*(-1)^n) - 1)/3);
%o A257852 A257852list(up_to) = { my(v = vector(up_to), i=0); for(a=1,oo, for(col=1,a, i++; if(i > up_to, return(v)); v[i] = A257852sq((a-(col-1)),col))); (v); };
%o A257852 v257852 = A257852list(up_to);
%o A257852 A257852(n) = v257852[n]; \\ _Antti Karttunen_, Apr 18 2024
%Y A257852 Cf. A006370, A075677, A096773 (after its initial 0, column 1 of this array).
%Y A257852 Cf. A004767, A017077, A082285, A238477 (rows 1-4).
%Y A257852 Cf. A371092, A371093 (column and row indices for odd numbers).
%Y A257852 Cf. also arrays A371095, A371096, A371097, A371100, A371101, A371102, A371103.
%K A257852 nonn,tabl
%O A257852 1,1
%A A257852 _L. Edson Jeffery_, Jul 12 2015