A228539 Rows of binary Walsh matrices interpreted as reverse binary numbers.

0, 0, 2, 0, 10, 12, 6, 0, 170, 204, 102, 240, 90, 60, 150, 0, 43690, 52428, 26214, 61680, 23130, 15420, 38550, 65280, 21930, 13260, 39270, 4080, 42330, 49980, 27030, 0, 2863311530, 3435973836, 1717986918, 4042322160, 1515870810, 1010580540

Offset

0,3

Comments

T(n,k) is row k of the binary Walsh matrix of size 2^n read as reverse binary number. The left digit is always 0, so all entries are even.

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 A228540.)

Divisibility by Fermat numbers:

All entries 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, Binary Walsh matrix of size 256

Formula

T(n,k) + A228540(n,k) = 2^2^n - 1

T(n,2^n-1) = A122570(n+1)

Example

Binary Walsh matrix of size 4 and row 2 of the triangle:
0 0 0 0         0
0 1 0 1        10
0 0 1 1        12
0 1 1 0         6
Triangle starts:
   k  =  0     1     2     3     4     5     6     7     8     9    10    11 ...
n
0        0
1        0     2
2        0    10    12     6
3        0   170   204   102   240    90    60   150
4        0 43690 52428 26214 61680 23130 15420 38550 65280 21930 13260 39270 ...

Crossrefs

Cf. A228540 (the same for the negated binary Walsh matrix).

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

Sequence in context: A303350 A351879 A070681 * A061189 A019220 A019140

Adjacent sequences: A228536 A228537 A228538 * A228540 A228541 A228542

Keyword

nonn,tabf

Author

Tilman Piesk, Aug 24 2013

Status

approved