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A228539 Rows of binary Walsh matrices interpreted as reverse binary numbers. 3
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 (list; graph; refs; listen; history; text; internal format)
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.

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 ...

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.)

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

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.

CROSSREFS

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

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

Sequence in context: A213322 A151887 A070681 * A061189 A019220 A019140

Adjacent sequences:  A228536 A228537 A228538 * A228540 A228541 A228542

KEYWORD

nonn,tabf

AUTHOR

Tilman Piesk, Aug 24 2013

STATUS

approved

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Last modified December 5 19:19 EST 2016. Contains 278770 sequences.