A320042 - OEIS

%I #38 Sep 01 2020 13:41:20

%S 0,-1,1,0,-2,0,2,1,-1,-3,-1,1,-1,1,3,2,0,-2,0,-2,-4,-2,0,2,0,-2,0,2,0,

%T 2,4,3,1,-1,1,-1,-3,-1,1,-1,-3,-5,-3,-1,-3,-1,1,3,1,-1,1,-1,-3,-1,1,3,

%U 1,-1,1,3,1,3,5,4,2,0,2,0,-2,0,2,0,-2,-4,-2,0,-2,0,2,0

%N a(n) = a(floor(n/2)) + (-1)^(n*(n+1)/2) with a(1)=0.

%C For 2^(2*k-1) - 1 < n < 2^(2*k), k>0, there's no n such that a(n)=0.

%C For 2^(2*k) - 1 < n < 2^(2*k+1), k>0, there are A000984(k+1) n's such that a(n)=0.

%H Robert Israel, <a href="/A320042/b320042.txt">Table of n, a(n) for n = 1..10000</a>

%F a(1) = 0, a(n) = a(floor(n/2)) + (-1)^(n*(n+1)/2).

%F a(n) = 2*A092339(n+1) - A000523(n).

%e a(9) = a(4) + (-1)^45 = -1, a(10) = a(5) + (-1)^55 = -3.

%e For 7 < n < 16, there's no n such that a(n)=0; for 15 < n < 32, there are 6 n's such that a(n)=0.

%p a:=proc(n) `if`(n=1,0,a(floor(n/2))+(-1)^(n*(n+1)/2)) end: seq(a(n),n=1..100); # _Muniru A Asiru_, Oct 07 2018

%t a[1] = 0; a[n_] := a[n] = a[Floor[n/2]] + (-1)^(n*(n + 1)/2); Table[a@n, {n, 1, 50}]

%o (PARI) a(n) = if (n==1, 0, a(n\2) + (-1)^(n*(n+1)/2)); \\ _Michel Marcus_, Oct 05 2018

%Y Cf. A000523, A000984, A007088, A092339.

%K sign,look,hear

%O 1,5

%A _Jinyuan Wang_, Oct 03 2018

%E More terms from _Michel Marcus_, Oct 05 2018