0,1

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 c*alpha^n - b/(alpha - 1) <= f_n <= c*alpha^n - a/(alpha - 1).

Here alpha = 4/3, a = -1/3, and b = 1/3, so we conclude that c*(4/3)^n - 1 < a(n) = f_n < c*(4/3)^n + 1 for some constant c, since we have neither e_n = -1/3 for all sufficiently large n nor e_n = 1/3 for all sufficiently large n.

Paolo Xausa, Table of n, a(n) for n = 0..5000

a(n) = A087192(n+1) + 1. - Jianing Song, Dec 27 2025

NestList[Round[4*#/3] &, 2, 50] (* Paolo Xausa, Nov 19 2025 *)

(PARI) A390254_up_to_N(N) = my(v = vector(N+1)); v[1] = 2; for(n=1, N, v[1+n] = round(4*v[n]/3)); v \\ gives a(0..N)

Cf. A087192.

Limit of a(n)/(4/3)^n is given by A390321. Numbers m such that A390254(m) > (resp. <) A390321*(4/3)^m are given by A390315 (resp. A390316).

Cf. A061418, A061419.

Sequence in context: A272948 A164001 A117598 * A120149 A117597 A241336

Adjacent sequences: A390251 A390252 A390253 * A390255 A390256 A390257

nonn,easy

Jianing Song, Oct 30 2025

approved