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Rows of negated binary Walsh matrices interpreted as reverse binary numbers.
4

%I #22 Oct 28 2021 10:00:43

%S 1,3,1,15,5,3,9,255,85,51,153,15,165,195,105,65535,21845,13107,39321,

%T 3855,42405,50115,26985,255,43605,52275,26265,61455,23205,15555,38505,

%U 4294967295,1431655765,858993459,2576980377,252645135,2779096485,3284386755

%N Rows of negated binary Walsh matrices interpreted as reverse binary numbers.

%C 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.

%C 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.)

%C Divisibility by Fermat numbers:

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

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

%H Tilman Piesk, <a href="/A228540/b228540.txt">Rows 0..8 of the triangle, flattened</a>

%H Tilman Piesk, <a href="/A228540/a228540.txt">Prime factorizations</a>

%H Tilman Piesk, <a href="http://commons.wikimedia.org/wiki/File:Binary_Walsh_matrix_256_neg.svg">Negated binary Walsh matrix of size 256</a>

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

%F T(n,0) = A051179(n)

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

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

%e Negated binary Walsh matrix of size 4 and row 2 of the triangle:

%e 1 1 1 1 15

%e 1 0 1 0 5

%e 1 1 0 0 3

%e 1 0 0 1 9

%e Triangle starts:

%e k = 0 1 2 3 4 5 6 7 8 9 10 11 ...

%e n

%e 0 1

%e 1 3 1

%e 2 15 5 3 9

%e 3 255 85 51 153 15 165 195 105

%e 4 65535 21845 13107 39321 3855 42405 50115 26985 255 43605 52275 26265 ...

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

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

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

%K nonn,tabf

%O 0,2

%A _Tilman Piesk_, Aug 24 2013