

A215409


The Goodstein sequence G_n(3).


24




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, 3341, 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 basen 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(n3)), 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, #ni, B(#ni, 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(n3)) \\ 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 A153012 A275300
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



