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Replace 9^k with (-1)^k in base-9 expansion of n.
4

%I #14 Apr 23 2022 16:33:00

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

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

%U -2,-1,0,1,2,-7,-6,-5,-4,-3,-2,-1,0,1,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9,0,1,2,3,4,5,6,7,8,-1,0,1,2,3,4

%N Replace 9^k with (-1)^k in base-9 expansion of n.

%C If n has base-9 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 + 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).

%F a(n) = n + 10 * Sum_{k>=1} (-1)^k * floor(n/9^k).

%e 89 = 108_9, 8 - 0 + 1 = 9, so a(89) = 9.

%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 + 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]

%t Table[n + 10 Sum[(-1)^k Floor[n/9^k], {k, 1, Floor[Log[9, 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, 9)[1:][::-1]))

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

%Y Cf. A007095, A053830, A055017, A065359, A065368, A346688, A346689, A346690, A346691, A346731.

%K sign,base

%O 0,3

%A _Ilya Gutkovskiy_, Jul 30 2021