A346731 Replace 8^k with (-1)^k in base-8 expansion of n.

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

Offset: 0

Author

Ilya Gutkovskiy, Jul 30 2021

Keywords

• sign
• base

Comments

If n has base-8 expansion abc..xyz with least significant digit z, a(n) = z - y + x - w + ...

Examples

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

Crossrefs

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

Programs

Mathematica

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]
Table[n + 9 Sum[(-1)^k Floor[n/8^k], {k, 1, Floor[Log[8, n]]}], {n, 0, 104}]

Python

from sympy.ntheory.digits import digits
def a(n):
    return sum(bi*(-1)**k for k, bi in enumerate(digits(n, 8)[1:][::-1]))
print([a(n) for n in range(105)]) # Michael S. Branicky, Jul 31 2021

Formula

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).

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