A065368 Alternating sum of ternary digits in n. Replace 3^k with (-1)^k in ternary expansion of n.

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

Offset: 0

Author

Marc LeBrun, Oct 31 2001

Keywords

• base
• easy
• sign

Comments

Notation: (3)[n](-1).

Fixed point of the morphism 0 -> 0,1,2; 1 -> -1,0,1; 2 -> -2,-1,0; ...; n -> -n,-n+1,-n+2. - Philippe Deléham, Oct 22 2011

Examples

			15 = +1(9)+2(3)+0(1) -> +1(+1)+2(-1)+0(+1) = -1 = a(15).
		

Crossrefs

Cf. A030341, A053735, A065359, A065364.

Programs

Python

from sympy.ntheory.digits import digits
def a(n):
    return sum(bi*(-1)**k for k, bi in enumerate(digits(n, 3)[1:][::-1]))
print([a(n) for n in range(104)]) # Michael S. Branicky, Jul 28 2021

Python

from sympy.ntheory import digits
def A065368(n): return sum((0, 1, 2, -1, 0, 1, -2, -1, 0)[i] for i in digits(n,9)[1:]) # Chai Wah Wu, Jul 19 2024

Formula

a(n) = Sum_{k>=0} A030341(n,k)*(-1)^k. - Philippe Deléham, Oct 22 2011.

G.f. A(x) satisfies: A(x) = x * (1 + 2*x) / (1 - x^3) - (1 + x + x^2) * A(x^3). - Ilya Gutkovskiy, Jul 28 2021

Extensions

Initial 0 added by Philippe Deléham, Oct 22 2011