A360079 - OEIS

%I #16 Jan 01 2024 19:49:38

%S 1,-2,0,1,0,1,0,-1,1,0,0,-1,0,0,0,1,0,-2,0,0,0,0,0,1,0,0,-1,0,0,-1,0,

%T 0,0,0,0,1,0,0,0,1,0,-1,0,0,-1,0,0,0,-1,0,0,0,0,2,0,0,0,0,0,2,0,0,-1,

%U -1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-2

%N Finite differences of Moebius function for the floor quotient poset.

%C a(n) = mu(n) - mu(n-1), where mu(n) = A360078(n) is the Moebius function of the floor quotient poset.

%H Andrew Howroyd, <a href="/A360079/b360079.txt">Table of n, a(n) for n = 1..10000</a>

%H J.-P. Cardinal, <a href="https://arxiv.org/abs/0811.3701">Symmetric matrices related to the Mertens function</a>, arXiv:0811.3701 [math.NT], 2008.

%H J. C. Lagarias and D. H. Richman, <a href="https://arxiv.org/abs/2212.11689">The floor quotient partial order</a>, arXiv:2212.11689 [math.NT], 2022.

%t LinearSolve[Table[If[Floor[i/j] > Floor[i/(j + 1)], 1, 0], {i, n}, {j, n}] . Table[If[i >= j, 1, 0], {i, n}, {j, n}], UnitVector[n, 1]]

%o (PARI) seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, my(S=Set(vector(n-1, k, n\(k+1)))); v[n]=-sum(i=1, #S, v[S[i]])); vector(#v, i, v[i]-if(i>1, v[i-1]))} \\ _Andrew Howroyd_, Jan 24 2023

%Y Cf. A002321, A008683, A360078.

%K sign

%O 1,2

%A _Harry Richman_, Jan 24 2023