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A254067 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = S(4*A257499(n,k) - 3), n,k >= 1, where the function S is as defined in A257480.

%I A254067 #12 Nov 05 2015 14:31:10
%S A254067 1,8,4,5,17,7,68,32,26,10,41,149,59,35,13,608,284,230,86,44,16,365,
%T A254067 1337,527,311,113,53,19,5468,2552,2066,770,392,140,62,22,3281,12029,
%U A254067 4739,2795,1013,473,167,71,25,49208,22964,18590,6926,3524,1256,554,194,80,28
%N A254067 Rectangular array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = S(4*A257499(n,k) - 3), n,k >= 1, where the function S is as defined in A257480.
%C A254067 Theorem: For all indices n and k such that n + k > 2, log(A(n,k))/log(A257499(n,k)) < log_2(3).
%C A254067 Conjecture: Arranging the sequence in ascending order gives A189707 (positions of 0 in A189706).
%F A254067 A(n,k) = S(4*A257499(n,k) - 3) = (3 + 3^n*(6*k - 3 + 2*(-1)^n))/6, where the function S is as defined in A257480.
%F A254067 For all k, A(1,k) <= A257499(1,k), and A(n,k) > A257499(n,k), for all n > 1.
%e A254067 .       1      4      7     10     13     16     19     22     25     28
%e A254067 .       8     17     26     35     44     53     62     71     80     89
%e A254067 .       5     32     59     86    113    140    167    194    221    248
%e A254067 .      68    149    230    311    392    473    554    635    716    797
%e A254067 .      41    284    527    770   1013   1256   1499   1742   1985   2228
%e A254067 .     608   1337   2066   2795   3524   4253   4982   5711   6440   7169
%e A254067 .     365   2552   4739   6926   9113  11300  13487  15674  17861  20048
%e A254067 .    5468  12029  18590  25151  31712  38273  44834  51395  57956  64517
%e A254067 .    3281  22964  42647  62330  82013 101696 121379 141062 160745 180428
%e A254067 .   49208 108257 167306 226355 285404 344453 403502 462551 521600 580649
%t A254067 (* Array antidiagonals flattened: *)
%t A254067 v[x_] := IntegerExponent[x, 2]; f[x_] := (3*x + 1)/2^v[3*x + 1]; s[x_] := (3 + (3/2)^v[1 + f[x]] (1 + f[x]))/6; A257499[n_, k_] := (1 + 2^n*(6*k - 3 + 2*(-1)^n))/3; A254067[n_, k_] := s[4*A257499[n, k] - 3]; Flatten[Table[A254067[n - k + 1, k], {n, 10}, {k, n}]]
%K A254067 nonn,tabl
%O A254067 1,2
%A A254067 _L. Edson Jeffery_, May 02 2015