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A215409 The Goodstein sequence G(3). 5
3, 3, 3, 2, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

The first term in the Goodstein sequence G(m) is m itself. To get the 2nd term, write m in hereditary base 2 notation (see links), change all the 2s to 3s, and then subtract 1 from the result. To get the 3rd term, write the 2nd term in hereditary base 3 notation, change all 3s to 4s, and subtract 1 again. Continue until the result is zero (by Goodstein's Theorem), when the sequence terminates.

REFERENCES

R. Goodstein, On the restricted ordinal theorem, J. Symbolic Logic 9 (1944), 33-41.

LINKS

Table of n, a(n) for n=1..6.

Eric Weisstein's World of Mathematics, Hereditary Representation

Eric Weisstein's World of Mathematics, Goodstein Sequence

Eric Weisstein's World of Mathematics, Goodstein's Theorem

Wikipedia, Hereditary base-n notation

Wikipedia, Goodstein sequence

Wikipedia, Goodstein's Theorem

Reinhard Zumkeller, Haskell programs for Goodstein sequences

EXAMPLE

a(1) = 3 = 2^1 + 1; a(2) = 3^1 + 1 - 1 = 3^1 = 3; a(3) = 4^1 - 1 = 3; a(4) = 3 - 1 = 2; a(5) = 2 - 1 = 1; a(6) = 1 - 1 = 0.

PROG

(Haskell)  see Link

CROSSREFS

Cf. A056004, A056041, A056193, A057650, A059933, A059934, A059935, A059936.

Sequence in context: A076237 A201432 A128210 * A239232 A153012 A016651

Adjacent sequences:  A215406 A215407 A215408 * A215410 A215411 A215412

KEYWORD

nonn,fini,full

AUTHOR

Jonathan Sondow, Aug 10 2012

STATUS

approved

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Last modified December 19 16:32 EST 2014. Contains 252236 sequences.