A222117 Goodstein sequence starting with 15.

15, 111, 1283, 18752, 326593, 6588344, 150994943, 3524450280, 100077777775, 3138578427934, 106993479003783, 3937376861542204, 155568096352467863, 6568408356994335930, 295147905181357143919, 14063084452070776884879, 708235345355342213988445

Offset

0,1

Comments

To calculate a(n+1), write a(n) in the hereditary representation base n+2, then bump the base to n+3, then subtract 1;

Compare to A222113: the underlying variants to define Goodstein sequences are equivalent.

Links

Nicholas Matteo, Table of n, a(n) for n = 0..383 (first 249 terms from Reinhard Zumkeller)

R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic, Vol. 9, No. 2, Jun., 1944

Eric Weisstein's World of Mathematics, Goodstein Sequence

Wikipedia, Goodstein's Theorem

Reinhard Zumkeller, Haskell programs for Goodstein sequences

Example

The first terms are:
a(0) = 2^(2+1) + 2^2 + 2^1 + 2^0 = 15;
a(1) = 3^(3+1) + 3^3 + 3^1 + 3^0 - 1 = 111;
a(2) = 4^(4+1) + 4^4 + 4^1 - 1 = 4^(4+1) + 4^4 + 3*4^0 = 1283;
a(3) = 5^(5+1) + 5^5 + 3*5^0 - 1 = 5^(5+1) + 5^5 + 2*5^0 = 18752;
a(4) = 6^(6+1) + 6^6 + 2*6^0 - 1 = 6^(6+1) + 6^6 + 1 = 326593;
a(5) = 7^(7+1) + 7^7 + 1 - 1 = 6588344;
a(6) = 8^(8+1) + 8^8 - 1 = 150994943.

Prog

(Haskell) -- See Link
(PARI) lista(nn) = {print1(a = 15, ", "); 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, ", "); ); } \\ Michel Marcus, Feb 24 2016

Crossrefs

Cf. A215409 (start=3), A056193 (start=4), A059933 (start=16), A211378 (start=19).

Sequence in context: A254869 A034184 A092646 * A105051 A105040 A298123

Adjacent sequences: A222114 A222115 A222116 * A222118 A222119 A222120

Keyword

nonn,fini

Author

Reinhard Zumkeller, Feb 13 2013

Extensions

Offset changed to 0 by Nicholas Matteo, Aug 21 2019

Status

approved