login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A327580 Triangle read by rows: T(n,k) = Sum_{1<=j*k<=n} cos(Pi*(j*k-1)/2). 1
1, 1, 0, 0, 0, -1, 0, 0, -1, 0, 1, 0, -1, 0, 1, 1, 0, -1, 0, 1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 1, 1, 0, 0, 0, 1, 0, -1, 0, 1, 0, 0, 0, 0, 0, 1, 0, -1, 0, 1, 0, -1, 0, 0, 0, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, 0, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1

COMMENTS

a(n) is a (-1, 0, 1)-valued sequence.

The identity Sum_{k=1..n} T(n,k) mu(k)=1 holds for every n. The function mu is the Moebius function and T(n,k) is entry k of row n in the triangular array.

T(n,2k) = 0 for all k.

T(n,k) is 4-periodic when k > n/3.

T(n,*) gives the transpose of the n-th truncation of Dirichlet convolution by (1,1,1,...) applied to the periodic sequence (1,0,-1,0,...).

LINKS

Jeffery Kline, Table of n, a(n) for n = 1..4950 (Rows n = 1..99 of triangle, flattened)

Jeffery Kline, Unital sums of the Moebius and Mertens functions, Journal of Integer Sequences, 23 (2020), Article 20.8.1.

EXAMPLE

The first 20 rows of the triangle:

1;

1, 0;

0, 0, -1;

0, 0, -1, 0;

1, 0, -1, 0, 1;

1, 0, -1, 0, 1, 0;

0, 0, -1, 0, 1, 0, -1;

0, 0, -1, 0, 1, 0, -1, 0;

1, 0,  0, 0, 1, 0, -1, 0, 1;

1, 0,  0, 0, 1, 0, -1, 0, 1, 0;

0, 0,  0, 0, 1, 0, -1, 0, 1, 0, -1;

0, 0,  0, 0, 1, 0, -1, 0, 1, 0, -1, 0;

1, 0,  0, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1;

1, 0,  0, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0;

0, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1;

0, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0;

1, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1;

1, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0;

0, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1;

0, 0, -1, 0, 0, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0, 1, 0, -1, 0;

...

MAPLE

T:= (n, k)-> add(cos(Pi*(j*k-1)/2), j=1..n/k):

seq(seq(T(n, k), k=1..n), n=1..14);  # Alois P. Heinz, Sep 29 2019

PROG

(Python)

import numpy as np

for n in range(1, 21):

    A = [ sum([ np.cos(np.pi * (j*k-1)/2) for k in range(1, n//j+1)])

            for j in range(1, n+1) ]

    print(np.array(np.rint(A), dtype=int))

(Python)

import numpy as np

m, maxn = 1, 1000

D = np.zeros((maxn, maxn), dtype=int);

for j in range(maxn): D[j::j+1, j]=1

cm = np.zeros(maxn, dtype=int); cm[   ::4*m] =  1; cm[2*m::4*m] = -1

for n in range(1, 21): print( D[:n, :n].T.dot(cm[:n]))

CROSSREFS

Cf. A002321, A008683, A056594.

Sequence in context: A143064 A185124 A185125 * A163811 A163817 A266837

Adjacent sequences:  A327577 A327578 A327579 * A327581 A327582 A327583

KEYWORD

sign,tabl

AUTHOR

Jeffery Kline, Sep 17 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 18 23:16 EDT 2021. Contains 347548 sequences. (Running on oeis4.)