cp's OEIS Frontend

A228540 Rows of negated binary Walsh matrices interpreted as reverse binary numbers.

%I A228540 #22 Oct 28 2021 10:00:43
%S A228540 1,3,1,15,5,3,9,255,85,51,153,15,165,195,105,65535,21845,13107,39321,
%T A228540 3855,42405,50115,26985,255,43605,52275,26265,61455,23205,15555,38505,
%U A228540 4294967295,1431655765,858993459,2576980377,252645135,2779096485,3284386755
%N A228540 Rows of negated binary Walsh matrices interpreted as reverse binary numbers.
%C A228540 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 A228540 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 A228540 Divisibility by Fermat numbers:
%C A228540 All entries in rows n >= 1 are divisible by F_0 =  3, except those with k = 1.
%C A228540 All entries in rows n >= 3 are divisible by F_2 = 17, except those with k = 4..7.
%H A228540 Tilman Piesk, <a href="/A228540/b228540.txt">Rows 0..8 of the triangle, flattened</a>
%H A228540 Tilman Piesk, <a href="/A228540/a228540.txt">Prime factorizations</a>
%H A228540 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 A228540 T(n,k) + A228539(n,k) = 2^2^n - 1
%F A228540 T(n,0) = A051179(n)
%F A228540 T(n,2^n-1) = A122569(n+1)
%F A228540 A211344(n,k) = T(n,2^(n-k))
%e A228540 Negated binary Walsh matrix of size 4 and row 2 of the triangle:
%e A228540 1 1 1 1        15
%e A228540 1 0 1 0         5
%e A228540 1 1 0 0         3
%e A228540 1 0 0 1         9
%e A228540 Triangle starts:
%e A228540       k  =  0     1     2     3    4     5     6     7   8     9    10    11 ...
%e A228540 n
%e A228540 0           1
%e A228540 1           3     1
%e A228540 2          15     5     3     9
%e A228540 3         255    85    51   153   15   165   195   105
%e A228540 4       65535 21845 13107 39321 3855 42405 50115 26985 255 43605 52275 26265 ...
%Y A228540 A228539 (the same for the binary Walsh matrix, not negated)
%Y A228540 A197818 (antidiagonals of the negated binary Walsh matrix converted to decimal).
%Y A228540 A000215 (Fermat numbers), A023394 (Prime factors of Fermat numbers).
%K A228540 nonn,tabf
%O A228540 0,2
%A A228540 _Tilman Piesk_, Aug 24 2013