A002716
An infinite coprime sequence defined by recursion.
Original entry on oeis.org
3, 5, 13, 17, 241, 257, 65281, 65537, 4294901761, 4294967297, 18446744069414584321, 18446744073709551617, 340282366920938463444927863358058659841
Offset: 0
- 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).
- A. W. F. Edwards, Infinite coprime sequences, Math. Gaz., 48 (1964), 416-422.
- A. W. F. Edwards, Infinite coprime sequences, Math. Gaz., 48 (1964), 416-422. [Annotated scanned copy]
- 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], 2012-2018.
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a[0] = 3; a[1] = 5;
a[n_] := a[n] = If[OddQ[n], a[n-1] + a[n-2] - 1, a[n-1]^2 - 3*a[n-1] + 3];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Aug 16 2018, after Michel Somos *)
-
{a(n) = if( n<2, 3 * (n>=0) + 2 * (n>0), if( n%2, a(n-1) + a(n-2) - 1, a(n-1)^2 - 3 * a(n-1) + 3))} /* Michael Somos, Feb 01 2004 */
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
-
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
-
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)))}
A001510
a(n) = 2*a(n-1)*(a(n-1)-1) for n > 1, with a(0) = 1, a(1) = 2.
Original entry on oeis.org
1, 2, 4, 24, 1104, 2435424, 11862575248704, 281441383062305809756861824, 158418504200047111075388369241884118003210485743490304
Offset: 0
- 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).
- A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
- A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
- H. E. Salzer, The approximation of numbers as sums of reciprocals, Amer. Math. Monthly, Vol. 54, No. 3 (1947), pp. 135-142.
- Index entries for sequences of form a(n+1)=a(n)^2 + ...
-
(* a5 = A002715 *) a5[n_?OddQ] := a5[n] = 2*a5[n-1] + 1; a5[n_?EvenQ] := a5[n] = (a5[n-1]^2 - 3)/2; a5[0] = 3; a[n_] := a5[2*n - 4] + 1; a[0] = 1; a[1] = 2; Table[a[n], {n, 0, 8}] (* Jean-François Alcover, Jan 25 2013, after R. J. Mathar *)
Join[{1}, RecurrenceTable[{a[1] == 2, a[n] == 2*a[n - 1]*(a[n - 1] - 1)}, a, {n, 1, 8}]] (* Amiram Eldar, Feb 02 2022 *)
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