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.

1, 8, 4, 5, 17, 7, 68, 32, 26, 10, 41, 149, 59, 35, 13, 608, 284, 230, 86, 44, 16, 365, 1337, 527, 311, 113, 53, 19, 5468, 2552, 2066, 770, 392, 140, 62, 22, 3281, 12029, 4739, 2795, 1013, 473, 167, 71, 25, 49208, 22964, 18590, 6926, 3524, 1256, 554, 194, 80, 28

Offset: 1

Author

L. Edson Jeffery, May 02 2015

Keywords

• nonn
• tabl

Comments

Theorem: For all indices n and k such that n + k > 2, log(A(n,k))/log(A257499(n,k)) < log_2(3).

Conjecture: Arranging the sequence in ascending order gives A189707 (positions of 0 in A189706).

Examples

			.       1      4      7     10     13     16     19     22     25     28
.       8     17     26     35     44     53     62     71     80     89
.       5     32     59     86    113    140    167    194    221    248
.      68    149    230    311    392    473    554    635    716    797
.      41    284    527    770   1013   1256   1499   1742   1985   2228
.     608   1337   2066   2795   3524   4253   4982   5711   6440   7169
.     365   2552   4739   6926   9113  11300  13487  15674  17861  20048
.    5468  12029  18590  25151  31712  38273  44834  51395  57956  64517
.    3281  22964  42647  62330  82013 101696 121379 141062 160745 180428
.   49208 108257 167306 226355 285404 344453 403502 462551 521600 580649
		

Programs

Mathematica

(* Array antidiagonals flattened: *)
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}]]

Formula

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.

For all k, A(1,k) <= A257499(1,k), and A(n,k) > A257499(n,k), for all n > 1.