1,5

Only odd indices make sense. The given sequence is a(1), a(3), a(5), etc.

This should be related to the Coxeter transformations for the posets of diagonally symmetric paths in an n*n grid. - F. Chapoton, Jun 11 2010

Start from 1, 1, -2, -2, -4, -4, 8, 8, 16, 16, -32, -32, -64, -64, 128, ... which is A016116(n-1) with negative signs in blocks of 4, assuming offset 1. The Mobius transform of that sequence is b(n) = 1, 0, -3, -3, -5, -2, 7, 10, 18, 20, -33, -25, -65, -72, 135, 120, ... for n >= 1, and the current sequence is a(n) = b(2n-1)/(2n-1). - R. J. Mathar, Oct 29 2011

b(1)=1*1; b(3)=-1*3; ...; b(9)=2*9.

A := proc(n)

(-1)^binomial(floor((n+1)/2), 2) * 2^floor((n-1)/2) ;

end proc:

L := [seq(A(n), n=1..40)] ;

b := MOBIUS(L) ;

for i from 1 to nops(b) by 2 do

printf("%d, ", op(i, b)/i) ;

end do: # R. J. Mathar, Oct 29 2011

b[n_] := Sum[(-1)^Binomial[(d+1)/2, 2]*2^((d-1)/2)*MoebiusMu[n/d], {d, Divisors[n]}]/n;

a[n_] := b[2n - 1];

a /@ Range[35] (* Jean-François Alcover, Mar 16 2020 *)

(SageMath)

def suite(n):

return sum((-1)**binomial(((d+1)//2), 2) * 2**((d-1)//2) * moebius(n//d) for d in divisors(n)) // n

[suite(n) for n in range(1, 22, 2)]

Similar to A022553 and A131868

Also related to A178749. - F. Chapoton, Jun 11 2010

Sequence in context: A090905 A065956 A328078 * A060013 A092424 A167510

Adjacent sequences: A178735 A178736 A178737 * A178739 A178740 A178741

sign,uned

F. Chapoton, Jun 08 2010

I would like a more precise definition. - N. J. A. Sloane, Jun 08 2010

approved