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Replace 8^k with (-1)^k in base-8 expansion of n.
4

%I #12 Aug 01 2021 14:43:35

%S 0,1,2,3,4,5,6,7,-1,0,1,2,3,4,5,6,-2,-1,0,1,2,3,4,5,-3,-2,-1,0,1,2,3,

%T 4,-4,-3,-2,-1,0,1,2,3,-5,-4,-3,-2,-1,0,1,2,-6,-5,-4,-3,-2,-1,0,1,-7,

%U -6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,0,1,2,3,4,5,6,7,-1,0,1,2,3,4,5,6,-2,-1,0,1,2,3,4,5,-3,-2,-1,0,1,2,3,4,-4

%N Replace 8^k with (-1)^k in base-8 expansion of n.

%C If n has base-8 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 + 4*x^3 + 5*x^4 + 6*x^5 + 7*x^6) / (1 - x^8) - (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7) * A(x^8).

%F a(n) = n + 9 * Sum_{k>=1} (-1)^k * floor(n/8^k).

%e 79 = 117_8, 7 - 1 + 1 = 7, so a(79) = 7.

%t nmax = 104; A[_] = 0; Do[A[x_] = x (1 + 2 x + 3 x^2 + 4 x^3 + 5 x^4 + 6 x^5 + 7 x^6)/(1 - x^8) - (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7) A[x^8] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

%t Table[n + 9 Sum[(-1)^k Floor[n/8^k], {k, 1, Floor[Log[8, 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, 8)[1:][::-1]))

%o print([a(n) for n in range(105)]) # _Michael S. Branicky_, Jul 31 2021

%Y Cf. A007094, A053829, A055017, A065359, A065368, A346688, A346689, A346690, A346691, A346732.

%K sign,base

%O 0,3

%A _Ilya Gutkovskiy_, Jul 30 2021