A258967 a(1)=1, a(2)=2, a(3)=3, a(n) = ceiling(sqrt(a(n-1)*a(n-2)*a(n-3))), n>3.
1, 2, 3, 3, 5, 7, 11, 20, 40, 94, 275, 1017, 5128, 37871, 444415, 9290130, 395420005, 40404949540, 12183091294648, 13951642918891149, 82872169787001239679, 3753148776564192982863648, 2083123034674803589767277778237, 25454214863632278822694894280883452911
Offset: 1
Keywords
Examples
a(4) = ceiling(sqrt(1*2*3)) = 3; a(5) = ceiling(sqrt(2*3*3)) = 5; a(6) = ceiling(sqrt(3*3*5)) = 7.
Links
- Harvey P. Dale, Table of n, a(n) for n = 1..39
Programs
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Magma
I:=[1, 2, 3]; [n le 3 select I[n] else Ceiling(Sqrt(Self(n-1)*Self(n-2)*Self(n-3))): n in [1..23]];
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Mathematica
RecurrenceTable[{a[n] == Ceiling[Sqrt[a[n - 1] a[n - 2] a[n - 3]]], a[1] == 1, a[2] == 2, a[3] == 3}, a, {n, 1, 23}] (* Michael De Vlieger, Jul 02 2015 *) a[1] = 1; a[2] = 2; a[3] = 3; a[n_] := a[n] = Ceiling[ Sqrt[ a[n - 1]*a[n - 2]*a[n - 3]]]; Array[a, 23] (* Robert G. Wilson v, Aug 12 2015 *) nxt[{a_,b_,c_}]:={b,c,Ceiling[Sqrt[a*b*c]]}; NestList[nxt,{1,2,3},30][[All,1]] (* Harvey P. Dale, Sep 09 2021 *) -
PARI
first(m)={my(v=vector(m));v[1]=1;v[2]=2;v[3]=3;for(i=4,m,v[i]=ceil(sqrt(v[i-1]*v[i-2]*v[i-3])));v;} \\ Anders Hellström, Aug 20 2015
Formula
a(n) is approximately k^(c^n), where c is the real root of x^3 - (x^2 + x + 1)/2 = 0 equal to (1 + (64 - 3*sqrt(417))^(1/3) + (64 + 3*sqrt(417))^(1/3))/6, and k is approximately 1.7450496...
Extensions
Corrected and extended by Harvey P. Dale, Sep 09 2021