A122570 -id:A122570 - cp's OEIS frontend

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

Tilman Piesk, Aug 24 2013

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.

			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 ...

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

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

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

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

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

A320916 Consider A010060 as a 2-adic number ...100110010110, then a(n) is its approximation up to 2^n.

Original entry on oeis.org

0, 0, 2, 6, 6, 22, 22, 22, 150, 406, 406, 406, 2454, 2454, 10646, 27030, 27030, 92566, 92566, 92566, 616854, 616854, 2714006, 6908310, 6908310, 6908310, 40462742, 107571606, 107571606, 376007062, 376007062, 376007062, 2523490710, 6818458006, 6818458006, 6818458006

Offset: 0

Jianing Song, Oct 26 2018

This is another interpretation of A010060 as a number, in a different way as considering it as a binary number.

Consider the g.f. of A010060. As a real-valued (or complex-valued) function it only converges for |x| < 1. In 2-adic field it only converges for |x|_2 < 1 as well, but here |x|_2 is a different metric. For a 2-adic number x, |x|_2 < 1 iff x is an even 2-adic integer.

			a(1) =     0_2 =  0.
a(2) =    10_2 =  2.
a(3) =   110_2 =  6.
a(4) =  0110_2 =  6.
a(5) = 10110_2 = 22.
...

Paolo Xausa, Table of n, a(n) for n = 0..1000

Cf. A010060, A122570, A019300 (bit reversal).

With[{nmax = 50}, Table[FromDigits[#[[-n;;]], 2], {n, 0, nmax}] & [ThueMorse[Range[nmax, 0, -1]]]] (* or *)
A320916[n_] := FromDigits[ThueMorse[Range[n-1, 0, -1]], 2]; Array[A320916, 51, 0] (* Paolo Xausa, Oct 18 2024 *)

a(n) = sum(i=0, n-1, 2^i*(hammingweight(i)%2))

a(n) = Sum_{i=0..n-1} A010060(i)*2^i (empty sum yields 0 for n = 0).

cp's OEIS Frontend

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

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