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

%I #20 Jan 15 2024 11:04:36

%S 0,1,2,3,-4,-3,-2,-1,-8,-7,-6,-5,-12,-11,-10,-9,16,17,18,19,12,13,14,

%T 15,8,9,10,11,4,5,6,7,32,33,34,35,28,29,30,31,24,25,26,27,20,21,22,23,

%U 48,49,50,51,44,45,46,47,40,41,42,43,36,37,38,39,-64,-63,-62,-61,-68,-67,-66,-65,-72,-71,-70,-69

%N Replace 4^k with (-4)^k in base 4 expansion of n.

%C Base 4 representation for n converted from base -4 to base 10.

%H Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, <a href="https://arxiv.org/abs/2012.04625">Finding structure in sequences of real numbers via graph theory: a problem list</a>, arXiv:2012.04625 [math.CO], 2020-2021.

%F a(4*k+m) = -4*a(k)+m for 0 <= m < 4. - _Chai Wah Wu_, Jan 16 2020

%t f[n_Integer, b_Integer] := Block[{l = IntegerDigits[n]}, Sum[l[[ -i]]*(-b)^(i - 1), {i, 1, Length[l]}]]; a = Table[ FromDigits[ IntegerDigits[n, 4]], {n, 0, 80}]; b = {}; Do[b = Append[b, f[a[[n]], 4]], {n, 1, 80}]; b

%o (PARI) a(n) = subst(Pol(digits(n,4)), x, -4); \\ _Michel Marcus_, Jan 30 2019

%Y Cf. A007090, A007608, A053985, A065369, A073792, A073793, A073794, A073795, A073796 & A073835.

%Y Cf. A066321, A320283.

%K base,easy,sign

%O 0,3

%A _Robert G. Wilson v_, Aug 12 2002