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A001685
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a(0) = 1, a(1) = 2, a(2) = 3; for n >= 3, a(n) = a(n-2) + a(n-1)*Product_{i=1..n-3} a(i).
(Formerly M0740 N0278)
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11
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1, 2, 3, 5, 13, 83, 2503, 976253, 31601312113, 2560404986164794683, 202523113189037952478722304798003, 506227391211661106785411233681995783881012463859772443053
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OFFSET
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0,2
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COMMENTS
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From a continued fraction.
Every term is relatively prime to all others. - Michael Somos, Feb 01 2004
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REFERENCES
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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).
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LINKS
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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
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FORMULA
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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
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MATHEMATICA
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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 *)
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PROG
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(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 */
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CROSSREFS
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Cf. A003686, A064526, A109134.
Sequence in context: A056162 A265785 A326372 * A074691 A139589 A152114
Adjacent sequences: A001682 A001683 A001684 * A001686 A001687 A001688
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane
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EXTENSIONS
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Edited by N. J. A. Sloane, Jun 12 2006
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STATUS
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approved
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