A228540 Rows of negated binary Walsh matrices interpreted as reverse binary numbers.

1, 3, 1, 15, 5, 3, 9, 255, 85, 51, 153, 15, 165, 195, 105, 65535, 21845, 13107, 39321, 3855, 42405, 50115, 26985, 255, 43605, 52275, 26265, 61455, 23205, 15555, 38505, 4294967295, 1431655765, 858993459, 2576980377, 252645135, 2779096485, 3284386755

Offset: 0

Author

Tilman Piesk, Aug 24 2013

Keywords

• nonn
• tabf

Comments

T(n,k) is row k of the negated binary Walsh matrix of size 2^n read as reverse binary number. The left digit is always 1, so all entries are odd.

Most of these numbers are divisible by Fermat numbers (A000215): All entries in all rows beginning with row n are divisible by F_(n-1), except the entries 2^(n-1)...2^n-1. (This is the same in A228539.)

Divisibility by Fermat numbers:

All entries in rows n >= 1 are divisible by F_0 = 3, except those with k = 1.

All entries in rows n >= 3 are divisible by F_2 = 17, except those with k = 4..7.

Examples

			Negated binary Walsh matrix of size 4 and row 2 of the triangle:
1 1 1 1        15
1 0 1 0         5
1 1 0 0         3
1 0 0 1         9
Triangle starts:
      k  =  0     1     2     3    4     5     6     7   8     9    10    11 ...
n
0           1
1           3     1
2          15     5     3     9
3         255    85    51   153   15   165   195   105
4       65535 21845 13107 39321 3855 42405 50115 26985 255 43605 52275 26265 ...
		

Links

• Tilman Piesk, Rows 0..8 of the triangle, flattened
• Tilman Piesk, Prime factorizations
• Tilman Piesk, Negated binary Walsh matrix of size 256

Crossrefs

A228539 (the same for the binary Walsh matrix, not negated)

A197818 (antidiagonals of the negated binary Walsh matrix converted to decimal).

A000215 (Fermat numbers), A023394 (Prime factors of Fermat numbers).

Formula

T(n,k) + A228539(n,k) = 2^2^n - 1

T(n,0) = A051179(n)

T(n,2^n-1) = A122569(n+1)

A211344(n,k) = T(n,2^(n-k))