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A372237 a(0) = 4; to obtain a(k), write out the base-(2^k) expansion of a(k-1), bump to base 2^(k+1), then subtract 1.

%I A372237 #12 Apr 25 2024 12:28:23
%S A372237 4,15,26,49,96,191,318,573,1084,2107,4154,8249,16440,32823,65590,
%T A372237 131125,262196,524339,1048626,2097201,4194352,8388655,16777262,
%U A372237 33554477,67108908,134217771,268435498,536870953,1073741864,2147483687,4294967334,8589934629,17179869220
%N A372237 a(0) = 4; to obtain a(k), write out the base-(2^k) expansion of a(k-1), bump to base 2^(k+1), then subtract 1.
%C A372237 Applying to the proof of the usual Goodstein's theorem to the ordinal number omega^omega shows that: for no matter what initial value and no matter what increasing sequence of bases b(0), b(1), ... with b(0) >= 2, the (weak) Goodstein sequence eventually terminates with 0. Here b(k) = 2^(k+1).
%C A372237 Sequence terminates at a(2^(2^70+70) + 2^70 + 68) = 0.
%H A372237 Jianing Song, <a href="/A372237/b372237.txt">Table of n, a(n) for n = 0..1000</a>
%H A372237 Googology Wiki, <a href="http://googology.wikia.com/wiki/Goodstein_sequence">Goodstein sequence</a>.
%H A372237 Wikipedia, <a href="http://en.wikipedia.org/wiki/Goodstein&#39;s_theorem">Goodstein's Theorem</a>
%F A372237 a(k) = 2^(k+2) + 68 - k for 5 <= k <= 68. The base-(2^(k+1)) expansion of a(k) consists of two digits 2 and 68 - k.
%F A372237 a(k) = 2^(k+1) + 2^70 + 68 - k for 69 <= 2^70 + 68. The base-(2^(k+1)) expansion of a(k) consists of two digits 1 and 2^70 + 68 - k.
%F A372237 a(k) = 2^(2^70+70) + 2^70 + 68 - k for 2^70 + 69 <= k <= 2^(2^70+70) + 2^70 + 68. The base-(2^(k+1)) expansion of a(k) consists of a single digit  2^(2^70+70) + 2^70 + 68 - k.
%e A372237 a(0) = 100_2 = 4;
%e A372237 a(1) = 100_4 - 1 = 15 = 33_4;
%e A372237 a(2) = 33_8 - 1 = 26 = 32_8;
%e A372237 a(3) = 32_16 - 1 = 49 = 31_16;
%e A372237 a(4) = 31_32 - 1 = 96 = 30_32;
%e A372237 a(5) = 30_64 - 1 = 191 = (2,63)_64.
%o A372237 (PARI) A372237_first_N_terms(N) = my(v=vector(N+1)); v[1] = 4; for(i=1, N, v[i+1] = fromdigits(digits(v[i],2^i),2^(i+1))-1); v
%Y A372237 Cf. A056193, A266202.
%K A372237 nonn,easy,fini
%O A372237 0,1
%A A372237 _Jianing Song_, Apr 23 2024