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A271559 a(n) = G_n(12), where G is the Goodstein function defined in A266201.

%I A271559 #14 Jan 11 2020 15:57:47
%S A271559 12,107,1065,15685,280019,5764910,134217867,3486784574,100000000211,
%T A271559 3138428376974,106993205379371,3937376385699637,155568095557812625,
%U A271559 6568408355712891083,295147905179352826375,14063084452067724991593,708235345355337676358285,37589973457545958193356327
%N A271559 a(n) = G_n(12), where G is the Goodstein function defined in A266201.
%C A271559 Goodstein's theorem shows that such sequence converges to zero for any starting value.
%H A271559 Nicholas Matteo, <a href="/A271559/b271559.txt">Table of n, a(n) for n = 0..383</a>
%H A271559 R. L. Goodstein, <a href="http://www.jstor.org/stable/2268019">On the Restricted Ordinal Theorem</a>, The Journal of Symbolic Logic 9, no. 2 (1944), 33-41.
%e A271559 G_1(12) = B_2(12)-1 = B_2(2^(2+1)+2^2)-1 = 3^(3+1)+3^3-1 = 107;
%e A271559 G_2(12) = B_3(3^(3+1)+2*3^2+2*3+2)-1 = 4^(4+1)+2*4^2+2*4+2-1 = 1065;
%e A271559 G_3(12) = B_4(4^(4+1)+2*4^2+2*4+1)-1 = 5^(5+1)+2*5^2+2*5+1-1 = 15685;
%e A271559 G_4(12) = B_5(5^(5+1)+2*5^2+2*5)-1 = 6^(6+1)+2*6^2+2*6-1 = 280019;
%e A271559 G_5(12) = B_6(6^(6+1)+2*6^2+6+5)-1 = 7^(7+1)+2*7^2+7+5-1 = 5764910;
%e A271559 G_6(12) = B_7(7^(7+1)+2*7^2+7+4)-1 = 8^(8+1)+2*8^2+8+4-1 = 134217867;
%e A271559 G_7(12) = B_8(8^(8+1)+2*8^2+8+3)-1 = 9^(9+1)+2*9^2+9+3-1 = 3486784574.
%o A271559 (PARI) lista(nn) = {print1(a = 12, ", "); for (n=2, nn, pd = Pol(digits(a, n)); q = sum(k=0, poldegree(pd), if (c=polcoeff(pd, k), c*x^subst(Pol(digits(k, n)), x, n+1), 0)); a = subst(q, x, n+1) - 1; print1(a, ", "); ); }
%Y A271559 Cf. A056193: G_n(4), A059933: G_n(16), A211378: G_n(19), A215409: G_n(3), A222117: G_n(15), A266204: G_n(5), A266205: G_n(6), A271554: G_n(7), A271555: G_n(8), A271556: G_n(9), A271557: G_n(10), A271558: G_n(11), A266201: G_n(n).
%K A271559 nonn,fini
%O A271559 0,1
%A A271559 _Natan Arie Consigli_, Apr 11 2016