A327562 a(0) = a(1) = 1; for n > 1, a(n) = (a(n-1) + a(n-2)) / gcd(a(n-1), a(n-2)) if a(n-1) and a(n-2) are not coprime, otherwise a(n) = a(n-1) + a(n-2) + 1.

1, 1, 3, 5, 9, 15, 8, 24, 4, 7, 12, 20, 8, 7, 16, 24, 5, 30, 7, 38, 46, 42, 44, 43, 88, 132, 5, 138, 144, 47, 192, 240, 9, 83, 93, 177, 90, 89, 180, 270, 5, 55, 12, 68, 20, 22, 21, 44, 66, 5, 72, 78, 25, 104, 130, 9, 140, 150, 29, 180, 210, 13, 224, 238, 33, 272, 306

Offset: 0

Author

Ian Band, Sep 16 2019

Keywords

• nonn

Comments

The sequence increases rapidly after n = 92.

From roughly n = 480 to n = 600, the sequence increases relatively fast and is almost a perfect exponential function.

Redefining a(0) and a(1) can result in drastically different sequences.

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A133058.

Programs

Magma

a:=[1,1]; for n in [3..67] do  if Gcd(a[n-1],a[n-2]) ne 1 then Append(~a,(a[n-1]+a[n-2])/Gcd(a[n-1],a[n-2])); else Append(~a,a[n-1]+a[n-2]+1); end if; end for; a; // Marius A. Burtea, Sep 19 2019

Mathematica

a[0]=a[1]=1; a[n_] := a[n] = Block[{g = GCD[a[n-1], a[n-2]]}, If[g==1,
a[n-1] + a[n-2] + 1, (a[n-1] + a[n-2])/g]]; Array[a, 67, 0] (* Giovanni Resta, Sep 19 2019 *)
nxt[{a_,b_}]:={b,If[!CoprimeQ[a,b],(a+b)/GCD[a,b],a+b+1]}; NestList[nxt,{1,1},70][[;;,1]] (* Harvey P. Dale, Feb 14 2024 *)

Python

import math
def a(n): # Iteratively generates an array containing the first n terms of a(n), n should be greater than 2
    a1 = 1 # this will hold a(n-1), its initial value is a(1)
    a2 = 1 # this will hold a(n-2), its initial value is a(0)
    terms = [None] * n
    terms[0] = a2
    terms[1] = a1
    for i in range(2, n):
        gcdPrev2 = math.gcd(a1, a2)
        if(gcdPrev2 > 1):
            terms[i] = int((a1 + a2) / gcdPrev2)
        else:
            terms[i] = a1 + a2 + 1
        a2 = a1
        a1 = terms[i]
    return terms

Formula

a(0) = a(1) = 1; for n > 1, a(n) = (a(n-1) + a(n-2)) / gcd(a(n-1), a(n-2)) if gcd(a(n-1), a(n-2)) > 1, otherwise a(n) = a(n-1) + a(n-2) + 1.