A100472 - OEIS

%I #13 Apr 07 2023 17:15:44

%S 1,8,80,640,6560,52480,524800,4198400,43046720,344373760,3443737600,

%T 27549900800,282386483200,2259091865600,22590918656000,

%U 180727349248000,1853020188851840,14824161510814720,148241615108147200

%N Inverse modulo 2 modulo transform of 9^n.

%C 9^n may be retrieved as 9^n = Sum_{k=0..n} (binomial(n,k) mod 2)*a(k).

%H G. C. Greubel, <a href="/A100472/b100472.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..n} (-1)^A010060(n-k) * (binomial(n, k) mod 2) * 9^k.

%F a(n) = Sum_{k=0..n} A106400(n-k) * (binomial(n, k) mod 2) * 9^k. - _G. C. Greubel_, Apr 06 2023

%t A100472[n_]:= A100472[n]= Sum[(-1)^ThueMorse[n-j]*Mod[Binomial[n, j], 2]*9^j, {j,0,n}];

%t Table[A100472[n], {n,0,30}] (* _G. C. Greubel_, Apr 06 2023 *)

%o (Magma)

%o A106400:= func< n | 1 - 2*(&+Intseq(n, 2) mod(2)) >;

%o A100472:= func< n | (&+[A106400(n-j)*(Binomial(n,j) mod 2)*9^j: j in [0..n]]) >;

%o [A100472(n): n in [0..30]]; // _G. C. Greubel_, Apr 06 2023

%o (SageMath)

%o @CachedFunction

%o def A010060(n): return (bin(n).count('1')%2)

%o def A100472(n): return sum((-1)^A010060(n-k)*(binomial(n, k)%2)*9^k for k in range(n+1))

%o [A100472(n) for n in range(31)] # _G. C. Greubel_, Apr 06 2023

%Y Cf. A001019, A010060, A047999, A106400.

%K easy,nonn

%O 0,2

%A _Paul Barry_, Dec 06 2004