

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.


26



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
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OFFSET

0,1


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.


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), 3341.
Eric Weisstein's World of Mathematics, Goodstein Sequence.
Wikipedia, Goodstein's Theorem
Reinhard Zumkeller, Haskell programs for Goodstein sequences


EXAMPLE

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.


PROG

(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


CROSSREFS

Cf. A056041, A056004, A057650, A059934, A059935, A059936, A271977.
Cf. A215409, A266204, A271554, A222117, A059933, A211378.
Sequence in context: A046963 A022386 A059178 * A196672 A306611 A102203
Adjacent sequences: A056190 A056191 A056192 * A056194 A056195 A056196


KEYWORD

nonn,fini


AUTHOR

Henry Bottomley, Aug 02 2000


EXTENSIONS

Edited by N. J. A. Sloane, Mar 06 2006
Offset changed to 0 by Nicholas Matteo, Sep 04 2019


STATUS

approved



