A280665 Recursive 1-parameter sequence a(n) allowing calculation of the Möbius function.

1, 0, 0, -1, 0, 0, 0, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 0, 1, 1, -1, 1, -1, -2, 3, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 1, 1, 1, -2, -1, 0, -1, 3, -1, 1, -1, 0, 1, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -2, 0, 0, 2, 2, -3, -1, 0, 0, 1, 1, -2, 2, -1, 1, -1

Offset

1,19

Comments

This sequence is generated from A266378 by excluding the second recursion parameter.

Links

Table of n, a(n) for n=1..117.

Formula

l(n) = floor((1/3)*(81+81*n+3*sqrt(729*n^2+1458*n+1104))^(1/3)-5/(81+81*n+3*sqrt(729*n^2+1458*n+1104))^(1/3))

c(n) = n*(n^2+3*n+8)/6 = A003600(n)

K(n) = n - 1 - c(l(n) - 1)

T(n,m) are coefficients of A008302

p(n) = c(l(n)-2)

u(n) = a(p(n)+K(n)+1)

v(n) = a(p(n)+K(n)-l(n)+2)

x(n) = a(p(n)+l(n)-1)*T(l(n)-1,l(n)*(l(n)-1)/2-K(n))

a(1) = 1

a(2) = 0

if (l(n)-2 >= K(n) or (1/2)*l(n)*(l(n)-1) < K(n)) then a(n) = 0 else a(n) = u(n)-v(n)-x(n)

Möbius(n) = a(c(n-1)+n)

A100198(n-2) = a(c(n-1)-n), for n>3.

Example

Möbius(2) = a(c(1)+2) and because the c(1)=2 => a(c(1)+2)= a(4). l(4)=2, K(4)=1 so l(4)-2<K(4) and l(4)*(l(4)-1)/2>=K(4) and a(4)=u(4)-v(4)-x(4)
p(4)=c(l(4)-2)=c(0)=0
u(4)=a(p(4)+K(4)+1)=a(2)=0
v(4)=a(p(4)+K(4)-l(4)+2)=a(1)=1
x(4)=a(p(4)+l(4)-1)*T(l(4)-1,l(4)*(l(4)-1)/2-K(4))=a(1)*T(1,0)=0, as T(1,0)=0.
a(4)=u(4)-v(4)-x(4)=0-1-0=-1.

Maple

l := n->floor((1/3)*(81+81*n+3*sqrt(1104+1458*n+729*n^2))^(1/3)-5/(81+81*n+3*sqrt(1104+1458*n+729*n^2))^(1/3)):
c := n->(1/6)*n*(n^2+3*n+8):
K := n->n-1-c(l(n)-1):
A := (n, z)->z*(product(z^i-1, i = 1 .. n-1)):
T := (n, k)->coeff(eval(A(n, z)), z, k):
p := n->c(l(n)-2):
u := n->a(p(n)+K(n)+1):
v := n->a(p(n)+K(n)-l(n)+2):
x := n->a(p(n)+l(n)-1)*T(l(n)-1, (1/2)*l(n)*(l(n)-1)-K(n)):
a := proc (n) option remember; if K(n) <= l(n)-2 or (1/2)*l(n)*(l(n)-1) < K(n) then 0 else u(n)-v(n)-x(n) end if end proc:
a(2) := 0:
a(1) := 1:

Crossrefs

Cf. A002321, A003600, A008302, A266378.

Sequence in context: A258772 A178401 A357911 * A356996 A037860 A037878

Adjacent sequences: A280662 A280663 A280664 * A280666 A280667 A280668

Keyword

sign

Author

Gevorg Hmayakyan, Jan 06 2017

Status

approved