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

Author

Reinhard Zumkeller, Feb 13 2013

Keywords

• nonn
• fini

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.

Examples

			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.
		

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

Crossrefs

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

Programs

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

Extensions

Offset changed to 0 by Nicholas Matteo, Aug 21 2019