OFFSET
0,4
COMMENTS
Like Narayana's Cows sequence A000930, except that the sums are divided by the greatest common divisor (gcd) of the prior terms.
It is a strong conjecture that 8 and 10 are missing from this sequence, but it would be nice to have a proof! See A214321 for the conjectured values. [I have often referred to this as "Reed Kelly's sequence" in talks.] - N. J. A. Sloane, Feb 18 2017
LINKS
T. D. Noe and N. J. A. Sloane, Table of n, a(n) for n = 0..10000
Benoit Cloitre, Graph of a(n)^(1/n) for n=1 up to 381817
N. J. A. Sloane, Exciting Number Sequences (video of talk), Mar 05 2021.
FORMULA
It appears that, very roughly, a(n) ~ constant*exp(0.123...*n). - N. J. A. Sloane, Sep 07 2012. See next comment for more precise estimate.
If a(n)^(1/n) converges the limit should be near 1.126 (see link). - Benoit Cloitre, Nov 08 2015
Robert G. Wilson v reports that at around 10^7 terms a(n)^(1/n) is about exp(1/8.4). - N. J. A. Sloane, May 05 2021
EXAMPLE
a(14)=9, a(16)=3, therefore a(17)=(9+3)/gcd(9,3) = 12/3 = 4.
a(24)=28, a(26)=60, therefore a(27)=(28+60)/gcd(28,60) = 88/4 = 22.
MAPLE
a:= proc(n) a(n):= `if`(n<3, 1, (a(n-1)+a(n-3))/igcd(a(n-1), a(n-3))) end:
seq(a(n), n=0..100); # Alois P. Heinz, Oct 18 2012
MATHEMATICA
t = {1, 1, 1}; Do[AppendTo[t, (t[[-1]] + t[[-3]])/GCD[t[[-1]], t[[-3]]]], {100}]
f[l_List] := Append[l, (l[[-1]] + l[[-3]])/GCD[l[[-1]], l[[-3]]]]; Nest[f, {1, 1, 1}, 62] (* Robert G. Wilson v, Jul 23 2012 *)
RecurrenceTable[{a[0]==a[1]==a[2]==1, a[n]==(a[n-1]+a[n-3])/GCD[ a[n-1], a[n-3]]}, a, {n, 70}] (* Harvey P. Dale, May 06 2014 *)
PROG
(Perl)
use bignum;
my @seq = (1, 1, 1);
print "1 1\n2 1\n3 1\n";
for ( my $i = 3; $i < 400; $i++ )
{
my $next = ( $seq[$i-1] + $seq[$i-3] ) /
gcd( $seq[$i-1], $seq[$i-3] );
my $ind = $i+1;
print "$ind $next\n";
push( @seq, $next );
}
sub gcd {
my ($x, $y) = @_;
($x, $y) = ($y, $x % $y) while $y;
return $x;
}
(Haskell)
a214551 n = a214551_list !! n
a214551_list = 1 : 1 : 1 : zipWith f a214551_list (drop 2 a214551_list)
where f u v = (u + v) `div` gcd u v
-- Reinhard Zumkeller, Jul 23 2012
(Sage)
def A214551Rec():
x, y, z = 1, 1, 1
yield x
while True:
x, y, z = y, z, (z + x)//gcd(z, x)
yield x
A214551 = A214551Rec();
print([next(A214551) for _ in range(65)]) # Peter Luschny, Oct 18 2012
(PARI) first(n)=my(v=vector(n+1)); for(i=1, min(n, 3), v[i]=1); for(i=4, #v, v[i]=(v[i-1]+v[i-3])/gcd(v[n-1], v[i-3])); v \\ Charles R Greathouse IV, Jun 21 2017
(Python)
from math import gcd
def aupton(nn):
alst = [1, 1, 1]
for n in range(3, nn+1):
alst.append((alst[n-1] + alst[n-3])//gcd(alst[n-1], alst[n-3]))
return alst
print(aupton(64)) # Michael S. Branicky, Mar 28 2022
CROSSREFS
KEYWORD
nonn,nice
AUTHOR
Reed Kelly, Jul 20 2012
STATUS
approved