OFFSET
0,3
COMMENTS
If n has base-9 expansion abc..xyz with least significant digit z, a(n) = z - y + x - w + ...
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 + 8*x^7) / (1 - x^9) - (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8) * A(x^9).
a(n) = n + 10 * Sum_{k>=1} (-1)^k * floor(n/9^k).
EXAMPLE
89 = 108_9, 8 - 0 + 1 = 9, so a(89) = 9.
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 + 8 x^7)/(1 - x^9) - (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8) A[x^9] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[n + 10 Sum[(-1)^k Floor[n/9^k], {k, 1, Floor[Log[9, n]]}], {n, 0, 104}]
PROG
(Python)
from sympy.ntheory.digits import digits
def a(n):
return sum(bi*(-1)**k for k, bi in enumerate(digits(n, 9)[1:][::-1]))
print([a(n) for n in range(105)]) # Michael S. Branicky, Jul 31 2021
CROSSREFS
KEYWORD
sign,base
AUTHOR
Ilya Gutkovskiy, Jul 30 2021
STATUS
approved