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A100472 Inverse modulo 2 modulo transform of 9^n.

%I A100472 #13 Apr 07 2023 17:15:44
%S A100472 1,8,80,640,6560,52480,524800,4198400,43046720,344373760,3443737600,
%T A100472 27549900800,282386483200,2259091865600,22590918656000,
%U A100472 180727349248000,1853020188851840,14824161510814720,148241615108147200
%N A100472 Inverse modulo 2 modulo transform of 9^n.
%C A100472 9^n may be retrieved as 9^n = Sum_{k=0..n} (binomial(n,k) mod 2)*a(k).
%H A100472 G. C. Greubel, <a href="/A100472/b100472.txt">Table of n, a(n) for n = 0..1000</a>
%F A100472 a(n) = Sum_{k=0..n} (-1)^A010060(n-k) * (binomial(n, k) mod 2) * 9^k.
%F A100472 a(n) = Sum_{k=0..n} A106400(n-k) * (binomial(n, k) mod 2) * 9^k. - _G. C. Greubel_, Apr 06 2023
%t A100472 A100472[n_]:= A100472[n]= Sum[(-1)^ThueMorse[n-j]*Mod[Binomial[n, j], 2]*9^j, {j,0,n}];
%t A100472 Table[A100472[n], {n,0,30}] (* _G. C. Greubel_, Apr 06 2023 *)
%o A100472 (Magma)
%o A100472 A106400:= func< n | 1 - 2*(&+Intseq(n, 2) mod(2)) >;
%o A100472 A100472:= func< n | (&+[A106400(n-j)*(Binomial(n,j) mod 2)*9^j: j in [0..n]]) >;
%o A100472 [A100472(n): n in [0..30]]; // _G. C. Greubel_, Apr 06 2023
%o A100472 (SageMath)
%o A100472 @CachedFunction
%o A100472 def A010060(n): return (bin(n).count('1')%2)
%o A100472 def A100472(n): return sum((-1)^A010060(n-k)*(binomial(n, k)%2)*9^k for k in range(n+1))
%o A100472 [A100472(n) for n in range(31)] # _G. C. Greubel_, Apr 06 2023
%Y A100472 Cf. A001019, A010060, A047999, A106400.
%K A100472 easy,nonn
%O A100472 0,2
%A A100472 _Paul Barry_, Dec 06 2004