OFFSET
0,2
COMMENTS
From a continued fraction.
Every term is relatively prime to all others. - Michael Somos, Feb 01 2004
REFERENCES
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
John Cerkan, Table of n, a(n) for n = 0..16
V. C. Harris, Another proof of the infinitude of primes, Amer. Math. Monthly, 63 (1956), 711.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - From N. J. A. Sloane, Jun 13 2012
FORMULA
a(n) = a(n-2) + a(n-1)*a(n-3)*(a(n-1)-a(n-3))/a(n-2). - Vaclav Kotesovec, May 21 2015
a(n) ~ c^(d^n), where d = A109134 = 1.754877666246692760049508896358528691894606617772793143989283970646... is the root of the equation d*(d-1)^2 = 1, c = 1.3081335128180696870655208993764956995000211962454918672885690026423582299... . - Vaclav Kotesovec, May 21 2015
EXAMPLE
For n=5, a(5) = 5 + 13 * (3*2*1) = 5 + 78 = 83. - Michael B. Porter, Nov 29 2025
MATHEMATICA
Clear[a]; a[0]=1; a[1]=2; a[2]=3; a[n_]:=a[n] = a[n-2] + a[n-1]*Product[a[j], {j, 1, n-3}]; Table[a[n], {n, 0, 15}] (* Vaclav Kotesovec, May 21 2015 *)
Clear[a]; RecurrenceTable[{a[n]==a[n-2]+a[n-1]*a[n-3]*(a[n-1]-a[n-3])/a[n-2], a[0]==1, a[1]==2, a[2]==3}, a, {n, 0, 15}] (* Vaclav Kotesovec, May 21 2015 *)
PROG
(PARI) a(n)=if(n<3, max(0, n+1), a(n-2)+a(n-1)*prod(i=1, n-3, a(i))) /* Michael Somos, Feb 01 2004 */
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Edited by N. J. A. Sloane, Jun 12 2006
STATUS
approved
