

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.


LINKS



FORMULA

a(0) = a(1) = a(2) = 3; a(3) = 2; a(4) = 1; a(n) = 0, n > 4;
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



KEYWORD



AUTHOR



EXTENSIONS



STATUS

approved



