A390321 - OEIS

%I #7 Nov 19 2025 15:59:19

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

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

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

%N Limit of A390254(n)/(4/3)^n.

%C In general, let {f_n}_{n>=0} be the sequence defined by f_{n+1} = alpha*f_n + e_n, where alpha > 1, a <= e_n <= b, then f_n = (f_0 + e_0/alpha + ... + e_{n-1}/alpha^n)*alpha^n = c*alpha^n - (e_n/alpha + e_{n+1}/alpha^2 + ...), where c = f_0 + Sum_{n>=0} e_n/alpha^{n+1}. We conclude that -(b/(alpha - 1))/alpha^n <= f_n/alpha^n - c <= -(a/(alpha - 1))/alpha^n.

%C Here we have alpha = 4/3, a = -1/3, and b = 1/3, so we have -(3/4)^n < A390254(n)/(4/3)^n - c < (3/4)^n, since we have neither e_n = -1/3 for all sufficiently large n nor e_n = 1/3 for all sufficiently large n.

%e 2.0763957307996667138140717653627546067142...

%o (PARI) A390321(prec) = my(N = ceil(log(2)/log(4/3) + prec*log(10)/log(4/3)), A390254_N = 2); for(n=1, N, A390254_N = round(4*A390254_N/3)); [(A390254_N-1)/(4/3.)^N, (A390254_N+1)/(4/3.)^N] \\ outputs a lower bound and an upper bound that differ by at most 10^(-prec), given sufficiently large realprecision

%Y Cf. A390254, A390315, A390316.

%Y Cf. A083286.

%K nonn

%O 1,1

%A _Jianing Song_, Nov 01 2025