A346688 - OEIS

%I #11 Aug 01 2021 14:43:51

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

%T 0,1,2,3,4,5,1,2,3,4,0,1,2,3,-1,0,1,2,3,4,5,6,2,3,4,5,1,2,3,4,0,1,2,3,

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

%N Replace 4^k with (-1)^k in base-4 expansion of n.

%C If n has base-4 expansion abc..xyz with least significant digit z, a(n) = z - y + x - w + ...

%F G.f. A(x) satisfies: A(x) = x * (1 + 2*x + 3*x^2) / (1 - x^4) - (1 + x + x^2 + x^3) * A(x^4).

%F a(n) = n + 5 * Sum_{k>=1} (-1)^k * floor(n/4^k).

%e 54 = 312_4, 2 - 1 + 3 = 4, so a(54) = 4.

%t nmax = 104; A[_] = 0; Do[A[x_] = x (1 + 2 x + 3 x^2)/(1 - x^4) - (1 + x + x^2 + x^3) A[x^4] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%t Table[n + 5 Sum[(-1)^k Floor[n/4^k], {k, 1, Floor[Log[4, n]]}], {n, 0, 104}]

%o (Python)

%o from sympy.ntheory.digits import digits

%o def a(n):

%o     return sum(bi*(-1)**k for k, bi in enumerate(digits(n, 4)[1:][::-1]))

%o print([a(n) for n in range(105)]) # _Michael S. Branicky_, Jul 29 2021

%Y Cf. A007090, A053737, A055017, A065359, A065368, A346689, A346690, A346691.

%K sign,base

%O 0,3

%A _Ilya Gutkovskiy_, Jul 29 2021