This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A137411 #31 Aug 15 2025 09:12:09
%S A137411 11,30,67,127,217,343,511,636,775,928,1095,1276,1471,1680,1903,2139,
%T A137411 2389,2653,2931,3223,3529,3849,4183,4531,4893,5269,5659,6063,6481,
%U A137411 6913,7359,7818,8291,8778,9279,9794,10323,10866,11423,11994,12579,13178
%N A137411 Weak Goodstein sequence starting at 11.
%C A137411 The sequence eventually goes to zero, as can be seen by noting that multiples of the highest exponent (3 in this case) only go down; in fact the 8th term, a(8) = 7*8^2 + 7*8 + 7 = 511; after which the multiple of the square term will only go down, etc.
%C A137411 This sequence, for 11, grows beyond the quintillions of digits before going to zero.
%C A137411 From _Zhuorui He_, Aug 07 2025: (Start)
%C A137411 For more info see A266201-A266202.
%C A137411 This sequence has A266203(11)+1 terms and a(A266203(11))=0 is the last term of this sequence. The maximum term in this sequence is a((A266203(11)-1)/2)=(A266203(11)+1)/2. 10^^8 < A266203(11) < 10^^9.
%C A137411 More precisely, 10^(10^(10^(10^(10^(10^(10^619.29937)))))) < A266203(11) < 10^(10^(10^(10^(10^(10^(10^619.299371)))))). (End)
%D A137411 K. Hrbacek and T. Jech, Introduction to Set Theory, Taylor & Francis Group, 1999, pp. 125-127.
%H A137411 Zhuorui He, <a href="/A137411/b137411.txt">Table of n, a(n) for n = 0..10000</a> (Terms 0..998 from Harvey P. Dale)
%F A137411 To obtain a(n + 1), write a(n) in base n + 2, increase the base to n + 3 and subtract 1.
%e A137411 a(0) = 11 = 2^3 + 2^1 + 2^0
%e A137411 a(1) = 3^3 + 3^1 + 3^0 - 1 = 30
%e A137411 a(2) = 4^3 + 4^1 - 1 = 4^3 + 3*4^0 = 67
%t A137411 nxt[{n_,a_}]:={n+1,FromDigits[IntegerDigits[a,n+1],n+2]-1}; Transpose[ NestList[ nxt,{1,11},50]][[2]] (* _Harvey P. Dale_, Feb 09 2015 *)
%o A137411 (PARI) a(n, m=11) = { my(wn = m); for (k=2, n+1, wn = fromdigits(digits(wn, k), k+1) - 1); wn; } \\ _Zhuorui He_, Aug 08 2025
%Y A137411 Cf. A056004 (strong Goodstein sequences), A059933 (strong Goodstein sequence for 16.).
%Y A137411 Weak Goodstein sequences: A267647: g_n(4); A267648: g_n(5); A271987: g_n(6); A271988: g_n(7); A271989: g_n(8); A271990: g_n(9); A271991: g_n(10); A137411: g_n(11); A271992: g_n(16); A265034: g_n(266); A266202: g_n(n); A266203: a(n)=k such that g_k(n)=0;
%K A137411 nonn,fini
%O A137411 0,1
%A A137411 Nicholas Matteo (kundor(AT)kundor.org), Apr 15 2008
%E A137411 Offset changed to 0 by _Zhuorui He_, Aug 07 2025