A258415 Array A read by upward antidiagonals in which the entry in row n and column k is defined by A(n,k) = (2 + 2^(n-1)*(6*k - 3 + 2*(-1)^n))/3, n,k >= 1.

1, 4, 3, 2, 8, 5, 14, 10, 12, 7, 6, 30, 18, 16, 9, 54, 38, 46, 26, 20, 11, 22, 118, 70, 62, 34, 24, 13, 214, 150, 182, 102, 78, 42, 28, 15, 86, 470, 278, 246, 134, 94, 50, 32, 17, 854, 598, 726, 406, 310, 166, 110, 58, 36, 19

Offset: 1

Author

L. Edson Jeffery, May 29 2015

Keywords

• nonn
• tabl

Comments

The sequence is a permutation of the natural numbers.

Theorem: Let v(y) denote the 2-adic valuation of y. For x an odd natural number, let F(x) = (3*x+1)/2^v(3*x+1) (see A075677). Row n of A is the set of all natural numbers m such that v(1+F(4*(2*m-1)-3)) = n.

Examples

			Array begins:
.      1     3     5     7     9    11    13    15    17     19
.      4     8    12    16    20    24    28    32    36     40
.      2    10    18    26    34    42    50    58    66     74
.     14    30    46    62    78    94   110   126   142    158
.      6    38    70   102   134   166   198   230   262    294
.     54   118   182   246   310   374   438   502   566    630
.     22   150   278   406   534   662   790   918  1046   1174
.    214   470   726   982  1238  1494  1750  2006  2262   2518
.     86   598  1110  1622  2134  2646  3158  3670  4182   4694
.    854  1878  2902  3926  4950  5974  6998  8022  9046  10070
		

Crossrefs

Cf. A005408, A008586, A017089 (rows 1-3).

Cf. A075677, A257480, A257499.

Programs

Mathematica

(* Array: *)
Grid[Table[(2 + 2^(n - 1)*(6*k - 3 + 2*(-1)^n))/3, {n, 10}, {k, 10}]]
(* Array antidiagonals flattened: *)
Flatten[Table[(2 + 2^(n - k)*(6*k - 3 + 2*(-1)^(n - k + 1)))/3, {n, 10}, {k, n}]]

Formula

A(n,k) = (1 + A257499(n,k))/2.