A056193 Goodstein sequence starting with 4: to calculate a(n+1), write a(n) in the hereditary representation in base n+2, then bump the base to n+3, then subtract 1.

4, 26, 41, 60, 83, 109, 139, 173, 211, 253, 299, 348, 401, 458, 519, 584, 653, 726, 803, 884, 969, 1058, 1151, 1222, 1295, 1370, 1447, 1526, 1607, 1690, 1775, 1862, 1951, 2042, 2135, 2230, 2327, 2426, 2527, 2630, 2735, 2842, 2951, 3062, 3175, 3290, 3407

Offset: 0

Author

Henry Bottomley, Aug 02 2000

Keywords

• nonn
• fini

Comments

Goodstein's theorem shows that such a sequence converges to zero for any starting value [e.g. if a(0)=1 then a(1)=0; if a(0)=2 then a(3)=0; and if a(0)=3 then a(5)=0]. With a(0)=4 we have a(3*2^(3*2^27 + 27) - 3)=0, which is well beyond the 10^(10^8)-th term.

The second half of such sequences is declining and the previous quarter is stable.

The resulting sequence 0,1,3,5,3*2^402653211 - 3, ... (see Comments in A056041) grows too rapidly to have its own entry.

Examples

			a(0) = 4 = 2^2,
a(1) = 3^3 - 1 = 26 = 2*3^2 + 2*3 + 2,
a(2) = 2*4^2 + 2*4 + 2 - 1 = 41 = 2*4^2 + 2*4 + 1,
a(3) = 2*5^2 + 2*5 + 1 - 1 = 60 = 2*5^2 + 2*5,
a(4) = 2*6^2 + 2*6 - 1 = 83 = 2*6^2 + 6 + 5,
a(5) = 2*7^2 + 7 + 5 - 1 = 109 etc.
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000 (final 2 terms from Nicholas Matteo)
• R. L. Goodstein, On the Restricted Ordinal Theorem, The Journal of Symbolic Logic, Vol. 9, No. 2 (1944), 33-41.
• Eric Weisstein's World of Mathematics, Goodstein Sequence.
• Wikipedia, Goodstein's Theorem
• Reinhard Zumkeller, Haskell programs for Goodstein sequences

Crossrefs

Cf. A056041, A056004, A057650, A059934, A059935, A059936, A271977.

Cf. A215409, A266204, A271554, A222117, A059933, A211378.

Programs

Haskell

See Zumkeller link

PARI

lista(nn) = {print1(a = 4, ", "); 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 22 2016

Extensions

Edited by N. J. A. Sloane, Mar 06 2006

Offset changed to 0 by Nicholas Matteo, Sep 04 2019