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

%I A280665 #28 Jan 18 2017 23:08:02
%S A280665 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,
%T A280665 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,
%U A280665 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
%N A280665 Recursive 1-parameter sequence a(n) allowing calculation of the Möbius function.
%C A280665 This sequence is generated from A266378 by excluding the second recursion parameter.
%F A280665 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))
%F A280665 c(n) = n*(n^2+3*n+8)/6 = A003600(n)
%F A280665 K(n) = n - 1 - c(l(n) - 1)
%F A280665 T(n,m) are coefficients of A008302
%F A280665 p(n) = c(l(n)-2)
%F A280665 u(n) = a(p(n)+K(n)+1)
%F A280665 v(n) = a(p(n)+K(n)-l(n)+2)
%F A280665 x(n) = a(p(n)+l(n)-1)*T(l(n)-1,l(n)*(l(n)-1)/2-K(n))
%F A280665 a(1) = 1
%F A280665 a(2) = 0
%F A280665 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)
%F A280665 Möbius(n) = a(c(n-1)+n)
%F A280665 A100198(n-2) = a(c(n-1)-n), for n>3.
%e A280665 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)
%e A280665 p(4)=c(l(4)-2)=c(0)=0
%e A280665 u(4)=a(p(4)+K(4)+1)=a(2)=0
%e A280665 v(4)=a(p(4)+K(4)-l(4)+2)=a(1)=1
%e A280665 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.
%e A280665 a(4)=u(4)-v(4)-x(4)=0-1-0=-1.
%p A280665 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)):
%p A280665 c := n->(1/6)*n*(n^2+3*n+8):
%p A280665 K := n->n-1-c(l(n)-1):
%p A280665 A := (n, z)->z*(product(z^i-1, i = 1 .. n-1)):
%p A280665 T := (n, k)->coeff(eval(A(n, z)), z, k):
%p A280665 p := n->c(l(n)-2):
%p A280665 u := n->a(p(n)+K(n)+1):
%p A280665 v := n->a(p(n)+K(n)-l(n)+2):
%p A280665 x := n->a(p(n)+l(n)-1)*T(l(n)-1, (1/2)*l(n)*(l(n)-1)-K(n)):
%p A280665 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:
%p A280665 a(2) := 0:
%p A280665 a(1) := 1:
%Y A280665 Cf. A002321, A003600, A008302, A266378.
%K A280665 sign
%O A280665 1,19
%A A280665 _Gevorg Hmayakyan_, Jan 06 2017