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 A215409 The Goodstein sequence G_n(3). 24
 3, 3, 3, 2, 1, 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. 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

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Last modified December 7 18:28 EST 2023. Contains 367660 sequences. (Running on oeis4.)