A054582 Array read by antidiagonals upwards: A(m,k) = 2^m * (2k+1), m,k >= 0.

1, 2, 3, 4, 6, 5, 8, 12, 10, 7, 16, 24, 20, 14, 9, 32, 48, 40, 28, 18, 11, 64, 96, 80, 56, 36, 22, 13, 128, 192, 160, 112, 72, 44, 26, 15, 256, 384, 320, 224, 144, 88, 52, 30, 17, 512, 768, 640, 448, 288, 176, 104, 60, 34, 19, 1024, 1536, 1280, 896, 576, 352, 208, 120

Offset

0,2

Comments

First column of array is powers of 2, first row is odd numbers, other cells are products of these two, so every positive integer appears exactly once. [Comment edited to match the definition. - L. Edson Jeffery, Jun 05 2015]

An analogous N X N <-> N bijection based, not on the binary, but on the Fibonacci number system, is given by the Wythoff array A035513.

As an array, this sequence (hence also A135764) is the dispersion of the even positive integers. For the definition of dispersion, see the link "Interspersions and Dispersions." The fractal sequence of this dispersion is A003602. - Clark Kimberling, Dec 03 2010

Links

Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened

Clark Kimberling, Interspersions and Dispersions.

Index entries for sequences that are permutations of the natural numbers

Formula

As a sequence, if n is a triangular number, then a(n)=a(n-A002024(n))+2, otherwise a(n)=2*a(n-A002024(n)-1).

a(n) = A075300(n-1)+1.

Recurrence for the sequence: if a(n-1)=2*k is even, then a(n)=k+A006519(2*k); if a(n-1)=2*k+1 is odd, then a(n)=2^(k+1), a(0)=1. - Philippe Deléham, Dec 13 2013

m = A(A001511(m)-1, A003602(m)-1), for each m in A000027. - L. Edson Jeffery, Nov 22 2015

The triangle is T(n, k) = A(n-k, k) = 2^(n-k)*(2*k+1), for n >= 0 and k = 0..n. - Wolfdieter Lang, Jan 30 2019

Example

Northwest corner of array A:
    1     3     5     7     9    11    13    15    17    19
    2     6    10    14    18    22    26    30    34    38
    4    12    20    28    36    44    52    60    68    76
    8    24    40    56    72    88   104   120   136   152
   16    48    80   112   144   176   208   240   272   304
   32    96   160   224   288   352   416   480   544   608
   64   192   320   448   576   704   832   960  1088  1216
  128   384   640   896  1152  1408  1664  1920  2176  2432
  256   768  1280  1792  2304  2816  3328  3840  4352  4864
  512  1536  2560  3584  4608  5632  6656  7680  8704  9728
[Array edited to match the definition. - L. Edson Jeffery, Jun 05 2015]
From Philippe Deléham, Dec 13 2013: (Start)
a(13-1)=20=2*10, so a(13)=10+A006519(20)=10+4=14.
a(3-1)=3=2*1+1, so a(3)=2^(1+1)=4. (End)
From Wolfdieter Lang, Jan 30 2019: (Start)
The triangle T begins:
   n\k   0    1    2   3   4   5   6   7  8  9 10 ...
   0:    1
   1:    2    3
   2:    4    6    5
   3:    8   12   10   7
   4:   16   24   20  14   9
   5:   32   48   40  28  18  11
   6:   64   96   80  56  36  22  13
   7:  128  192  160 112  72  44  26  15
   8:  256  384  320 224 144  88  52  30 17
   9:  512  768  640 448 288 176 104  60 34 19
  10: 1024 1536 1280 896 576 352 208 120 68 38 21
  ...
T(3, 2) = 2^1*(2*2+1) = 10. (End)

Mathematica

(* Array: *)
Grid[Table[2^m*(2*k + 1), {m, 0, 9}, {k, 0, 9}]] (* L. Edson Jeffery, Jun 05 2015 *)
(* Array antidiagonals flattened: *)
Flatten[Table[2^(m - k)*(2*k + 1), {m, 0, 9}, {k, 0, m}]] (* L. Edson Jeffery, Jun 05 2015 *)

Prog

(Haskell)
a054582 n k = a054582_tabl !! n !! k
a054582_row n = a054582_tabl !! n
a054582_tabl = iterate
   (\xs@(x:_) -> (2 * x) : zipWith (+) xs (iterate (`div` 2) (2 * x))) [1]
a054582_list = concat a054582_tabl
-- Reinhard Zumkeller, Jan 22 2013
(PARI) T(m, k)=(2*k+1)<<m \\ Charles R Greathouse IV, Jun 21 2017

Crossrefs

The sequence is a permutation of A000027.

Main diagonal is A014480; inverse permutation is A209268.

Cf. A001511, A002024, A003602, A006519, A025480, A035513, A075300, A135764.

Sequence in context: A348711 A080997 A151942 * A257797 A220347 A099884

Adjacent sequences: A054579 A054580 A054581 * A054583 A054584 A054585

Keyword

easy,nice,nonn,tabl

Author

Henry Bottomley, Apr 12 2000

Extensions

Offset corrected by Reinhard Zumkeller, Jan 22 2013

Status

approved