A271554 - OEIS

%I #18 Jan 11 2020 15:57:47

%S 7,30,259,3127,46657,823543,16777215,37665879,77777775,150051213,

%T 273624711,475842915,794655639,1281445305,2004318063,3051893870,

%U 4537630813,6604718946,9431578931,13238000758,18291957825,24917131658,33501182551,44504801406,58471578053,76038721330

%N a(n) = G_n(7), where G is the Goodstein function defined in A266201.

%H Nicholas Matteo, <a href="/A271554/b271554.txt">Table of n, a(n) for n = 0..10000</a>

%H 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.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Goodstein%27s_theorem#Goodstein_sequences">Goodstein sequence</a>

%e G_1(7) = B_2(7) - 1 = B[2](2^2 + 2 + 1) - 1 = 3^3 + 3 + 1 - 1 = 30;

%e G_2(7) = B_3(G_1(7)) - 1 = B[3](3^3 + 3) - 1 =  4^4 + 4 - 1 = 259;

%e G_3(7) = B_4(G_2(7)) - 1 = 5^5 + 3 - 1 = 3127;

%e G_4(7) = B_5(G_3(7)) - 1 = 6^6 + 2 - 1 = 46657;

%e G_5(7) = B_6(G_4(7)) - 1 = 7^7 + 1 - 1 = 823543;

%e G_6(7) = B_7(G_5(7)) - 1 = 8^8 - 1 = 16777215;

%e G_7(7) = B_8(G_6(7)) - 1 = 7*9^7 + 7*9^6 + 7*9^5 + 7*9^4 + 7*9^3 + 7*9^2 + 7*9 + 7 - 1 = 37665879.

%o (PARI) lista(nn) = {print1(a = 7, ", "); 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 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), A266201: G_n(n).

%K nonn,fini

%O 0,1

%A _Natan Arie Consigli_, Apr 10 2016