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A215409 The Goodstein sequence G_n(3). 20
3, 3, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

G_0(m) = m. 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. Sequence converges to 0 (by Goodstein's Theorem).

Decimal expansion of 33321/10000. - Natan Arie' Consigli, Jan 23 2015

LINKS

Table of n, a(n) for n=0..64.

R. L. Goodstein, On the Restricted Ordinal Theorem, J. Symb. Logic 9, 33-41, 1944.

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

FORMULA

a(0) = a(1) = a(2) = 3;

a(3) = 2;

a(4) = 1;

a(n) = 0, n>4;

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;

a(7) = 0;

Etc.

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,easy,cons

AUTHOR

Jonathan Sondow, Aug 10 2012

EXTENSIONS

Corrected by Natan Arie' Consigli, Jan 23 2015

STATUS

approved

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Last modified May 31 18:53 EDT 2016. Contains 273548 sequences.