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A228539 Rows of binary Walsh matrices interpreted as reverse binary numbers. 3

%I

%S 0,0,2,0,10,12,6,0,170,204,102,240,90,60,150,0,43690,52428,26214,

%T 61680,23130,15420,38550,65280,21930,13260,39270,4080,42330,49980,

%U 27030,0,2863311530,3435973836,1717986918,4042322160,1515870810,1010580540

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

%C 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.

%C Triangle starts:

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

%C n

%C 0 0

%C 1 0 2

%C 2 0 10 12 6

%C 3 0 170 204 102 240 90 60 150

%C 4 0 43690 52428 26214 61680 23130 15420 38550 65280 21930 13260 39270 ...

%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 A228540.)

%H Tilman Piesk, <a href="/A228539/b228539.txt">Rows 0..8 of the triangle, flattened</a>

%H Tilman Piesk, <a href="/A228539/a228539.txt">Prime factorizations</a>

%H Tilman Piesk, <a href="http://commons.wikimedia.org/wiki/File:Binary_Walsh_matrix_256.svg">Binary Walsh matrix of size 256</a>

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

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

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

%e 0 0 0 0 0

%e 0 1 0 1 10

%e 0 0 1 1 12

%e 0 1 1 0 6

%e Divisibility by Fermat numbers:

%e All entries are divisible by F_0 = 3, except those with k = 1.

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

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

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

%K nonn,tabf

%O 0,3

%A _Tilman Piesk_, Aug 24 2013

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Last modified October 22 00:52 EDT 2019. Contains 328315 sequences. (Running on oeis4.)