A215409 The Goodstein sequence G_n(3).

3, 3, 3, 2, 1, 0

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. Continue until the result is zero (by Goodstein's Theorem), when the sequence terminates.

Decimal expansion of 33321/100000. - Natan Arie Consigli, Jan 23 2015

Links

Table of n, a(n) for n=0..5.

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.

Mathematica

PadRight[CoefficientList[Series[3 + 3 x + 3 x^2 + 2 x^3 + x^4, {x, 0, 4}], x], 6] (* 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, A057650, A059934, A059935, A059936, A271977.

Cf. A056041, A056193, A266204, A271554, A222117, A059933, A211378.

Sequence in context: A076237 A201432 A128210 * A239232 A346624 A153012

Adjacent sequences: A215406 A215407 A215408 * A215410 A215411 A215412

Keyword

cons,easy,nonn,fini,full

Author

Jonathan Sondow, Aug 10 2012

Extensions

Corrected by Natan Arie Consigli, Jan 23 2015

Status

approved