A006695 - OEIS

%I M0838 #29 Jun 28 2023 08:22:24

%S 2,3,7,13,97,193,18817,37633,708158977,1416317953,1002978273411373057,

%T 2005956546822746113,2011930833870518011412817828051050497,

%U 4023861667741036022825635656102100993

%N a(2n)=2*a(2n-2)^2-1, a(2n+1)=2*a(2n)-1, a(0)=2.

%C An infinite coprime sequence defined by recursion.

%C Every term is relatively prime to all others. - _Michael Somos_, Feb 01 2004

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A006695/b006695.txt">Table of n, a(n) for n=0..19</a>

%H S. Kalpazidou et al., <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa55/aa5542.pdf">Lüroth-type alternating series representations for real numbers</a>, Acta Arithmetica, 55 (1990), 311-322.

%H Jeffrey Shallit, <a href="http://www.fq.math.ca/Scanned/31-1/shallit.pdf">Rational numbers with non-terminating, non-periodic modified Engel-type expansions</a>, Fib. Quart., 31 (1993), 37-40.

%H <a href="/index/El#Engel">Index entries for sequences related to Engel expansions</a>

%F a(2n) = A001075(2^n).

%t nxt[{n_,a_,b_}]:=If[OddQ[n],{n+1,b,2a^2-1},{n+1,b,2b-1}]; Transpose[ NestList[ nxt,{1,2,3},15]][[2]] (* _Harvey P. Dale_, Jun 22 2015 *)

%o (PARI) a(n)=if(n<1,2*(n==0),if(n%2,2*a(n-1)-1,2*a(n-2)^2-1))

%Y Cf. A001075, A001685, A002715, A003686, A064526.

%K nonn,easy,nice

%O 0,1

%A _N. J. A. Sloane_