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A215409 The Goodstein sequence G_n(3). 23
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 33321/100000. - 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;

From Iain Fox, Dec 12 2017: (Start)

G.f.: 3 + 3*x + 3*x^2 + 2*x^3 + x^4.

E.g.f.: 3 + 3*x + (3/2)*x^2 + (1/3)*x^3 + (1/24)*x^4.

a(n) = floor(2 - (4/Pi)*arctan(n-3)), n >= 0.

(End)

EXAMPLE

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

a(1) = 3^1 + 1 - 1 = 3^1 = 3;

a(2) = 4^1 - 1 = 3;

a(3) = 3 - 1 = 2;

a(4) = 2 - 1 = 1;

a(5) = 1 - 1 = 0;

a(6) = 0;

etc.

MATHEMATICA

PadRight[CoefficientList[Series[3 + 3 x + 3 x^2 + 2 x^3 + x^4, {x, 0, 4}], x], 105] (* Michael De Vlieger, Dec 12 2017 *)

PROG

(Haskell)  see Link

(PARI) B(n, b)=sum(i=1, #n=digits(n, b), n[i]*(b+1)^if(#n<b+i, #n-i, B(#n-i, b)))

a(n) = my(x=3); for(i=1, n, x=B(x, i+1)-1; if(x==0, break())); x \\ (uses definition of sequence) Iain Fox, Dec 13 2017

(PARI) first(n) = my(res = vector(n)); res[1] = res[2] = res[3] = 3; res[4] = 2; res[5] = 1; res; \\ Iain Fox, Dec 12 2017

(PARI) first(n) = Vec(3 + 3*x + 3*x^2 + 2*x^3 + x^4 + O(x^n)) \\ Iain Fox, Dec 12 2017

(PARI) a(n) = floor(2 - (4/Pi)*atan(n-3)) \\ Iain Fox, Dec 12 2017

CROSSREFS

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

Sequence in context: A076237 A201432 A128210 * A239232 A153012 A275300

Adjacent sequences:  A215406 A215407 A215408 * A215410 A215411 A215412

KEYWORD

base,cons,easy,nonn

AUTHOR

Jonathan Sondow, Aug 10 2012

EXTENSIONS

Corrected by Natan Arie' Consigli, Jan 23 2015

STATUS

approved

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Last modified July 17 16:38 EDT 2019. Contains 325107 sequences. (Running on oeis4.)