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

Author

Tilman Piesk, Aug 24 2013

Keywords

• nonn
• tabf

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.

Examples

			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 ...
		

Links

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

Crossrefs

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

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

Formula

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

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