%I #39 Dec 10 2021 11:18:14
%S 1,3,5,9,13,21,7,37,53,85,17,29,149,213,341,11,69,117,597,853,1365,25,
%T 45,277,469,2389,3413,5461,15,101,181,1109,1877,9557,13653,21845,33,
%U 61,405,725,4437,7509,38229,54613,87381,19,133,245,1621,2901,17749,30037
%N Array T(n,k) of odd Collatz preimages read by antidiagonals.
%C Every odd number occurs uniquely in this array. See A178414.
%H T. D. Noe, <a href="/A178415/b178415.txt">T(n,k) for n = 1..50, by antidiagonals</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Collatz_conjecture">Collatz conjecture</a>
%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%F From _Bob Selcoe_, Apr 09 2015 (Start):
%F T(n,k) = 4*T(n,k-1) + 1.
%F T(n,k) = T(1,k) + 2^(2k+1)*(n-1)/2 when n is odd;
%F T(n,k) = T(2,k) + 4^k*(n-2)/2 when n >= 2 and n is even. So equivalently:
%F T(n,k) = T(n-2,k) + 2^(2k+1) when n is odd; and
%F T(n,k) = T(n-2,k) + 4^k when n is even.
%F Let j be the n-th positive odd number coprime with 3. Then:
%F T(n,k) = (j*4^k - 1)/3 when j == 1 (mod 3); and
%F T(n,k) = (j*2^(2k-1) - 1)/3 when j == 2 (mod 3).
%F (End)
%F From _Wolfdieter Lang_, Sep 18 2021: (Start)
%F T(n, k) = ((3*n - 1)*4^k - 2)/6 if n is even, and ((3*n - 2)*4^k - 1)/3 if n is odd, for n >= 1 and k >= 1. Also for n = 0: -A007583(k-1), with A007583(-1) = 1/2, and for k = 0: A022998(n-1)/2, with A022998(-1) = -1.
%F O.g.f. for array T (with row n = 0 and column k = 0; z for rows and x for columns): G(z, x) = (1/(2*(1-x)*(1-4*x)*(1-z^2)^2)) * ((2*x-4)*z^3 + (3-5*x)*z^2 + 2*x*z + 3*x - 1). (End)
%e Array T begins:
%e . 1 5 21 85 341 1365 5461 21845 87381 349525
%e . 3 13 53 213 853 3413 13653 54613 218453 873813
%e . 9 37 149 597 2389 9557 38229 152917 611669 2446677
%e . 7 29 117 469 1877 7509 30037 120149 480597 1922389
%e . 17 69 277 1109 4437 17749 70997 283989 1135957 4543829
%e . 11 45 181 725 2901 11605 46421 185685 742741 2970965
%e . 25 101 405 1621 6485 25941 103765 415061 1660245 6640981
%e . 15 61 245 981 3925 15701 62805 251221 1004885 4019541
%e . 33 133 533 2133 8533 34133 136533 546133 2184533 8738133
%e . 19 77 309 1237 4949 19797 79189 316757 1267029 5068117
%e - _L. Edson Jeffery_, Mar 11 2015
%e From _Bob Selcoe_, Apr 09 2015 (Start):
%e n=5, j=13: T(5,3) = 277 = (13*4^3 - 1)/3;
%e n=6, j=17: T(6,4) = 725 = (17*2^7 - 1)/3.
%e (End)
%t t[n_,1] := t[n,1] = If[OddQ[n],4n-3,2n-1]; t[n_,k_] := t[n,k] = 4*t[n,k-1]+1; Flatten[Table[t[n-i+1,i], {n,20}, {i,n}]]
%Y Rows of array: -A007583(k-1) (n=0), A002450 (n=1), A072197(k-1) (n=2), A206374(n=3), A072261 (n=4), A323824 (n=5), A072262 (n=6), A330246 (n=7), A072201 (n=8), ...
%Y Columns of array: A022998(n-1)/2 (k=0), A178414 (k=1), ...
%Y Cf. A347834 (permuted rows of the array).
%K nonn,tabl,easy
%O 1,2
%A _T. D. Noe_, May 28 2010