A006695
a(2n)=2*a(2n-2)^2-1, a(2n+1)=2*a(2n)-1, a(0)=2.
Original entry on oeis.org
2, 3, 7, 13, 97, 193, 18817, 37633, 708158977, 1416317953, 1002978273411373057, 2005956546822746113, 2011930833870518011412817828051050497, 4023861667741036022825635656102100993
Offset: 0
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- T. D. Noe, Table of n, a(n) for n=0..19
- S. Kalpazidou et al., Lüroth-type alternating series representations for real numbers, Acta Arithmetica, 55 (1990), 311-322.
- Jeffrey Shallit, Rational numbers with non-terminating, non-periodic modified Engel-type expansions, Fib. Quart., 31 (1993), 37-40.
- Index entries for sequences related to Engel expansions
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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 *)
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a(n)=if(n<1,2*(n==0),if(n%2,2*a(n-1)-1,2*a(n-2)^2-1))
A066356
Numerator of sequence defined by recursion c(n) = 1 + c(n-2) / c(n-1), c(0) = 0, c(1) = 1.
Original entry on oeis.org
0, 1, 1, 2, 3, 7, 23, 167, 3925, 661271, 2609039723, 1728952269242533, 4516579101127820242349159, 7812958861560974806259705508894834509747, 35298563436210937269618773778802420542715366288238091341051372773
Offset: 0
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nxt[{a_,b_}]:={b,1+a/b}; NestList[nxt,{0,1},20][[All,1]]//Numerator (* Harvey P. Dale, Sep 26 2016 *)
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{a(n) = if( n<4, max(0, n) - (n>1), (2 * a(n-1) * a(n-2)^2 - a(n-1)^2 * a(n-4) - a(n-2)^3 * a(n-3)) / (a(n-2) - a(n-3) * a(n-4)))}
A081475
Consider the mapping f(x/y) = (x+y)/(2xy) where x/y is a reduced fraction. Beginning with x_0 = 1 and y_0 = 2, repeated application of this mapping produces a sequence of fractions x_n/y_n; a(n) is the n-th numerator.
Original entry on oeis.org
1, 3, 7, 31, 367, 21199, 15311887, 648309901711, 19853227652502777487, 25742087295488761786102488482959, 1022127038655087543344600484892552190865956757100687
Offset: 0
The n-th application of the mapping produces the fraction x_n/y_n from the fraction x_(n-1)/y_(n-1):
n=1: f(1/2) = (1+2)/(2*1*2) = 3/4 (so a(1)=3);
n=2: f(3/4) = (3+4)/(2*3*4) = 7/24 (so a(2)=7);
n=3: f(7/24) = (7+24)/(2*7*24) = 31/336 (so a(3)=31).
From _Jianing Song_, Oct 10 2021: (Start)
a(0) = 1;
a(1) = 1 + 2^1 = 3;
a(2) = 3 + 2^2*1 = 7;
a(3) = 7 + 2^3*1*3 = 31;
a(4) = 31 + 2^4*1*3*7 = 367;
a(5) = 367 + 2^5*1*3*7*31 = 21199. (End)
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a(n)=local(v); if(n<2,n>0,v=[1,2];for(k=2,n,v=[v[1]+v[2],2*v[1]*v[2]]); v[1])
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lista(n) = my(v=vector(n+1)); v[1]=1; if(n>=1, v[2]=3); for(k=2, n, v[k+1] = v[k] + 2*v[k-1]*(v[k]-v[k-1])); v \\ Jianing Song, Oct 10 2021
Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003
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