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

0,2

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.

Links

Tilman Piesk, Rows 0..8 of the triangle, flattened

Tilman Piesk, Prime factorizations

Tilman Piesk, Negated binary Walsh matrix of size 256

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))

Example

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 ...

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).

Sequence in context: A126454 A293558 A259841 * A144815 A065250 A092589

Adjacent sequences: A228537 A228538 A228539 * A228541 A228542 A228543

Keyword

nonn,tabf

Author

Tilman Piesk, Aug 24 2013

Status

approved