A166407 -id:A166407 - cp's OEIS frontend

A166408 a(n) = floor(A166407(n)/3).

-1, 0, 0, 1, -3, 1, 0, 2, 0, 1, 0, 3, -10, 0, 0, 3, 0, 2, 0, 4, 0, 1, 0, 5, -21, 2, 0, 4, 0, 3, 0, 2, 0, 1, 0, 7, 0, 0, 0, 5, -27, 3, 0, 6, 0, 2, 0, 8, 0, 0, 0, 5, 0, 3, 0, 8, 0, 2, 0, 10, -55, 2, 0, 5, 0, 5, 0, 2, 0, 3, 0, 10, 0, 0, 0, 7, 0, 4, 0, 10, 0, 1, 0, 11, -78, 2, 0, 2, 0, 5, 0, 8, 0, 2, 0

Offset: 0

Antti Karttunen, Oct 21 2009

sign

See the conjecture in A166407. If true, then a(i) = A166406(i)/A005408(i), whenever i is not in A166101.

A165951(n)=a(A102781(n)) for n>=2.

from sympy import floor, jacobi_symbol as J
def a(n):
    l=0
    m=0
    for i in range(1, 2*n + 2):
        if J(i, 2*n + 1)==-1: l+=i
        elif J(i, 2*n + 1)==1: m+=i
    return floor(3*((l - m)/(2*n + 1)))//3
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 12 2017

A166406 a(n) = A166405(n)-A166100(n).

Original entry on oeis.org

-1, 1, 0, 7, -27, 11, 0, 30, 0, 19, 0, 69, -250, 9, 0, 93, 0, 70, 0, 156, 0, 43, 0, 235, -1029, 102, 0, 220, 0, 177, 0, 126, 0, 67, 0, 497, 0, 50, 0, 395, -2187, 249, 0, 522, 0, 182, 0, 760, 0, 0, 0, 515, 0, 321, 0, 888, 0, 230, 0, 1190, -6655, 246, 0, 635, 0, 655, 0

Offset: 0

Antti Karttunen, Oct 21 2009, Oct 22 2009

sign

Zeros occur at (A166409(k)-1)/2. The negative terms occur at positions given by A046092 (see the comment at A166040).

Sum of those positive i <= 2n+1, for which J(i,2n+1)=-1 minus sum of those positive i <= 2n+1, for which J(i,2n+1)=+1. Here J(i,k) is the Jacobi symbol.

Antti Karttunen, Table of n, a(n) for n = 0..65535

A125615(n)=a(A102781(n)). Cf. A166100, A166407-A166409. The cases where a(i)/A005408(i) is not integer seem also to be given by A166101.

from sympy import jacobi_symbol as J
def a(n):
    l=0
    m=0
    for i in range(1, 2*n + 2):
        if J(i, 2*n + 1)==-1: l+=i
        elif J(i, 2*n + 1)==1: m+=i
    return l - m
print([a(n) for n in range(101)]) # Indranil Ghosh, Jun 12 2017

cp's OEIS Frontend

A166408 a(n) = floor(A166407(n)/3).

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