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A178415 Array T(n,k) of odd Collatz preimages read by antidiagonals.

%I A178415 #40 Feb 16 2025 08:33:12
%S A178415 1,3,5,9,13,21,7,37,53,85,17,29,149,213,341,11,69,117,597,853,1365,25,
%T A178415 45,277,469,2389,3413,5461,15,101,181,1109,1877,9557,13653,21845,33,
%U A178415 61,405,725,4437,7509,38229,54613,87381,19,133,245,1621,2901,17749,30037
%N A178415 Array T(n,k) of odd Collatz preimages read by antidiagonals.
%C A178415 Every odd number occurs uniquely in this array. See A178414.
%H A178415 T. D. Noe, <a href="/A178415/b178415.txt">T(n,k) for n = 1..50, by antidiagonals</a>
%H A178415 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>
%H A178415 Wikipedia, <a href="http://en.wikipedia.org/wiki/Collatz_conjecture">Collatz conjecture</a>
%H A178415 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%F A178415 From _Bob Selcoe_, Apr 09 2015 (Start):
%F A178415 T(n,k) = 4*T(n,k-1) + 1.
%F A178415 T(n,k) = T(1,k) + 2^(2k+1)*(n-1)/2 when n is odd;
%F A178415 T(n,k) = T(2,k) + 4^k*(n-2)/2 when n >= 2 and n is even. So equivalently:
%F A178415 T(n,k) = T(n-2,k) + 2^(2k+1) when n is odd; and
%F A178415 T(n,k) = T(n-2,k) + 4^k when n is even.
%F A178415 Let j be the n-th positive odd number coprime with 3. Then:
%F A178415 T(n,k) = (j*4^k - 1)/3 when j == 1 (mod 3); and
%F A178415 T(n,k) = (j*2^(2k-1) - 1)/3 when j == 2 (mod 3).
%F A178415 (End)
%F A178415 From _Wolfdieter Lang_, Sep 18 2021: (Start)
%F A178415 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 A178415 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 A178415 Array T begins:
%e A178415 .    1    5   21    85   341   1365    5461   21845    87381   349525
%e A178415 .    3   13   53   213   853   3413   13653   54613   218453   873813
%e A178415 .    9   37  149   597  2389   9557   38229  152917   611669  2446677
%e A178415 .    7   29  117   469  1877   7509   30037  120149   480597  1922389
%e A178415 .   17   69  277  1109  4437  17749   70997  283989  1135957  4543829
%e A178415 .   11   45  181   725  2901  11605   46421  185685   742741  2970965
%e A178415 .   25  101  405  1621  6485  25941  103765  415061  1660245  6640981
%e A178415 .   15   61  245   981  3925  15701   62805  251221  1004885  4019541
%e A178415 .   33  133  533  2133  8533  34133  136533  546133  2184533  8738133
%e A178415 .   19   77  309  1237  4949  19797   79189  316757  1267029  5068117
%e A178415 - _L. Edson Jeffery_, Mar 11 2015
%e A178415 From _Bob Selcoe_, Apr 09 2015 (Start):
%e A178415 n=5, j=13: T(5,3) = 277 = (13*4^3 - 1)/3;
%e A178415 n=6, j=17: T(6,4) = 725 = (17*2^7 - 1)/3.
%e A178415 (End)
%t A178415 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 A178415 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 A178415 Columns of array: A022998(n-1)/2 (k=0), A178414 (k=1), ...
%Y A178415 Cf. A347834 (permuted rows of the array).
%K A178415 nonn,tabl,easy
%O A178415 1,2
%A A178415 _T. D. Noe_, May 28 2010