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A214551 a(n) = (a(n-1) + a(n-3))/gcd(a(n-1), a(n-3)) with a(0) = a(1) = a(2) = 1. 26
1, 1, 1, 2, 3, 4, 3, 2, 3, 2, 2, 5, 7, 9, 14, 3, 4, 9, 4, 2, 11, 15, 17, 28, 43, 60, 22, 65, 25, 47, 112, 137, 184, 37, 174, 179, 216, 65, 244, 115, 36, 70, 37, 73, 143, 180, 253, 36, 6, 259, 295, 301, 80, 75, 376, 57, 44, 105, 54, 49, 22, 38, 87, 109, 147 (list; graph; refs; listen; history; text; internal format)
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. - 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

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

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();

[A214551.next() for i 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

CROSSREFS

Similar to A000930.

Cf. A214320, A214321, A214322, A214323 (gcd's), A219898 (records), A214324, A214325, A214330, A214331, A214809, A227836, A227837.

Starting with a(2) = 3 gives A214626. - Reinhard Zumkeller, Jul 23 2012

Sequence in context: A122453 A017849 A134536 * A211010 A131731 A255480

Adjacent sequences:  A214548 A214549 A214550 * A214552 A214553 A214554

KEYWORD

nonn,nice

AUTHOR

Reed Kelly, Jul 20 2012

STATUS

approved

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Last modified November 22 05:36 EST 2017. Contains 295076 sequences.