A001685 -id:A001685 - cp's OEIS frontend

A003686 Number of genealogical 1-2 rooted trees of height n.

1, 2, 3, 5, 11, 41, 371, 13901, 5033531, 69782910161, 351229174914190691, 24509789089655802510792656021, 8608552999157278575508415639286249242844899051

Offset: 1

Vsevolod F. Lev, c. 1998

Let u(n), v(n) be defined by u(1) = v(1) = 1, u(n+1) = u(n) + v(n) and v(n+1) = u(n)*v(n) for n >= 1; then a(n) = u(n) and A064847(n) = v(n). - Benoit Cloitre, Apr 01 2002 [Edited by Petros Hadjicostas, May 11 2020]
Consider the mapping f(a/b) = (a + b)/(a*b). Taking a = 1 and b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 1/1, 2/1, 3/2, 5/6, 11/30, ... The current sequence contains the numerators. - Amarnath Murthy, Mar 24 2003
An infinite coprime sequence defined by recursion. - Michael Somos, Mar 19 2004

D. Parisse, The Tower of Hanoi and the Stern-Brocot Array, Thesis, Munich, 1997.

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..19
Index entries for sequences related to Stern's sequences
Index entries for sequences related to rooted trees

Cf. A001622, A001685, A064526, A064847, A070231, A070233, A070234, A094303.

I:=[1,2]; [n le 2 select I[n] else Self(n-1)+Self(n-2)*(Self(n-1)-Self(n-2)): n in [1..14]]; // Vincenzo Librandi, Jul 19 2016

RecurrenceTable[{a[1]==1, a[2]==2, a[n]==a[n-1]+a[n-2](a[n-1]-a[n-2])}, a[n],{n,15}] (* Harvey P. Dale, Jul 27 2011 *)
Re[NestList[Re@#+(1+I Re@#)Im@#&, 1+I, 15]] (* Vladimir Reshetnikov, Jul 18 2016 *)

a(n) = local(an); if(n<1, 0, an=vector(max(2,n)); an[1]=1; an[2]=2; for(k=3, n, an[k]=an[k-1] - an[k-2]^2 + an[k-1]*an[k-2]); an[n])

Limit_{n -> infinity} a(n)^phi/A064847(n) = 1, where phi = (1 + sqrt(5))/2 is the golden ratio. - Benoit Cloitre, May 08 2002
Numerator of b(n), where b(n) = 1/numerator(b(n-1)) + 1/denominator(b(n-1)) for n >= 2 with b(1) = 1.
a(n+1) = a(n) + a(1)*a(2)*...*a(n-1) for n >= 2. Also a(n+1) = a(n) + a(n-1)*(a(n) - a(n-1)) for n >= 2. In both cases, we start with a(1) = 1 and a(2) = 2.
a(n) ~ c^(phi^n), where c = 1.22508584062304325811405322247537613534139348463831009881946422737141574647... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2015

Additional description from Andreas M. Hinz and Daniele Parisse

A064526 Define a pair of sequences by p(0) = 0, q(0) = p(1) = q(1) = 1, q(n+1) = p(n)*q(n-1), p(n+1) = q(n+1) + q(n) for n > 0; then a(n) = p(n) and A064183(n) = q(n).

Original entry on oeis.org

0, 1, 2, 3, 5, 13, 49, 529, 21121, 10369921, 213952189441, 2214253468601687041, 473721461635593679669210030081, 1048939288228833101089604217183056027094304481281

Offset: 0

Michael Somos, Oct 07 2001

Every nonzero term is relatively prime to all others (which proves that there are infinitely many primes). See A236394 for the primes that appear.

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. - _N. J. A. Sloane_, Jun 13 2012
Michael Somos and R. Haas, A linked pair of sequences implies the primes are infinite, Amer. Math. Monthly, 110(6) (2003), 539-540.

Cf. A001685, A003686, A064183, A064847, A070231, A070233, A070234, A094303.

See A236394 for the primes that are produced.

Flatten[{0,1, RecurrenceTable[{a[n]==(a[n-1]^2 + a[n-2]^2 - a[n-1]*a[n-2] * (1+a[n-2]))/(1-a[n-2]), a[2]==2, a[3]==3},a,{n,2,15}]}] (* Vaclav Kotesovec, May 21 2015 *)

{a(n) = local(v); if( n<3, max(0, n), v = [1,1]; for( k=3, n, v = [v[2], v[1] * (v[1] + v[2])]); v[1] + v[2])}

{a(n) = if( n<4, max(0, n), (a(n-1)^2 + a(n-2)^2 - a(n-1) * a(n-2) * (1 + a(n-2))) / (1 - a(n-2)))}

a(n) = (a(n-1)^2 + a(n-2)^2 - a(n-1) * a(n-2) * (1 + a(n-2))) / (1 - a(n-2)) for n >= 2.

a(n) ~ c^(phi^n), where c = 1.2364241784241086061606568429916822975882631646194967549068405592472125928485... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2015

A109134 Decimal expansion of Phi, the real root of the equation 1/x = (x-1)^2.

Original entry on oeis.org

1, 7, 5, 4, 8, 7, 7, 6, 6, 6, 2, 4, 6, 6, 9, 2, 7, 6, 0, 0, 4, 9, 5, 0, 8, 8, 9, 6, 3, 5, 8, 5, 2, 8, 6, 9, 1, 8, 9, 4, 6, 0, 6, 6, 1, 7, 7, 7, 2, 7, 9, 3, 1, 4, 3, 9, 8, 9, 2, 8, 3, 9, 7, 0, 6, 4, 6, 0, 8, 0, 6, 5, 5, 1, 2, 8, 0, 8, 1, 0, 9, 0, 7, 3, 8, 2, 2, 7, 0, 9, 2, 8, 4, 2, 2, 5, 0, 3, 0, 3, 6, 4, 8, 3, 7

Offset: 1

Lekraj Beedassy, Aug 17 2005

The silver number (A060006) is equal to Phi*(Phi-1).
Also Phi*(Phi-1) = 1/(Phi-1). - Richard R. Forberg, Oct 08 2014
Equations to which this is a root can also be written as: x = sqrt(x + sqrt(x)); x^2 - x - sqrt(x) = 0; or this form where n = 1: x = n + 1/sqrt(x). When n = 2 then the root is 2.618033988... = A104457 = 1 + A001622 or 1 + "Golden Ratio" called phi. - Richard R. Forberg, Oct 08 2014
Also equals the largest root (negated) of the Mandelbrot polynomial P_2(z) = 1+z*(1+z)^2. - Jean-François Alcover, Apr 16 2015
Suppose that r is a real number in the interval [3/2, 5/3). Let C(r) = (c(k)) be the sequence of coefficients in the Maclaurin series for 1/(Sum_{k>=0} floor((k+1)*r))(-x)^k). Conjectures: the limit L(r) of c(k+1)/c(k) as k -> oo exists, L(r) is discontinuous at 5/3 (cf. A279676), and the left limit of L(r) as r->5/3 is Phi. - Clark Kimberling, Jul 11 2017
From Wolfdieter Lang, Nov 07 2022: (Start)
This equals r + 2/3 where r is the real root of y^3 - (1/3)*y - 25/27.
The other roots of x^3 - 2*x^2 + x - 1 are (2 + w1*((25 + 3*sqrt(69))/2)^(1/3) + w2*((25 - 3*sqrt(69))/2)^(1/3))/3 = 0.1225611668... + 0.7448617668...*i, and its complex conjugate, where w1 = (-1 + sqrt(3)*i)/2 = exp(2*Pi*i/3) and w2 = (-1 - sqrt(3)*i)/2 are the complex conjugate roots of x^3 - 1.
Using hyperbolic functions these roots are (2 - cosh((1/3)*arccosh(25/2)) + sqrt(3)*sinh((1/3)*arccosh(25/2))*i)/3, and its complex conjugate. (End)

			1.75487766624669276004950889635852869189460661777279314398928397064...

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 5.11, p. 340.
Martin Gardner, A Gardner's Workout, pp. 124-126, A. K. Peters MA 2001.

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
Simon Baker, On small bases which admit countably many expansions, Journal of Number Theory, Volume 147, February 2015, Pages 515-532.
Simon Plouffe, Plouffe's Inverter .
Nikita Sidorov, Expansions in non-integer bases: Lower, middle and top orders, Journal of Number Theory, Volume 129, Issue 4, April 2009, Pages 741-754. See Prop. 2.3 p. 744.
Yuru Zou and Derong Kong, On a problem of countable expansions, Journal of Number Theory, Volume 158, January 2016, Pages 134-150. See Theorem 1.1 p. 135.
Index entries for algebraic numbers, degree 3.

Cf. A001622, A001685, A060006, A075778, A104457, A358181.

FindRoot[x^3 - 2x^2 + x - 1 == 0, {x, 1.75}, WorkingPrecision -> 128][[1, 2]] (* Robert G. Wilson v, Aug 19 2005 *)
Root[x^3-2x^2+x-1, x, 1] // RealDigits[#, 10, 105]& // First (* Jean-François Alcover, Mar 05 2013 *)

d=104;default(realprecision,d);print(k=solve(x=1,2,(x-1)^2-1/x)); for(c=0,d,z=floor(k);print1(z,",",);k=10*(k-z))

polrootsreal(x^3-2*x^2+x-1)[1] \\ Charles R Greathouse IV, Aug 15 2014

Equals 1+A075778. - R. J. Mathar, Aug 20 2008
Equals (1/6*(108+12*sqrt(69))^(1/3) + 2/(108+12*sqrt(69))^(1/3))^2. - Vaclav Kotesovec, Oct 08 2014
Equals Rho^2 where Rho is the plastic number 1.3247179572...(see A060006). - Philippe Deléham, Sep 29 2020
From Wolfdieter Lang, Nov 07 2022: (Start)
Equals (2 + ((25 + 3*sqrt(69))/2)^(1/3) + ((25 + 3*sqrt(69))/2)^(-1/3))/3.
Equals (2 + ((25 + 3*sqrt(69))/2)^(1/3) + ((25 - 3*sqrt(69))/2)^(1/3))/3.
Equals 2*(1 + cosh((1/3)*arccosh(25/2)))/3. (End)
Equals - Sum_{k>=1} Gamma(k - k/5 - 1)*Gamma(k/5 + 1)*sin(3*k*Pi/5)/(k*Pi*Gamma(k)). - Antonio Graciá Llorente, Dec 14 2024

Extended by Klaus Brockhaus and Robert G. Wilson v, Aug 19 2005

A002715 An infinite coprime sequence defined by recursion.

Original entry on oeis.org

3, 7, 23, 47, 1103, 2207, 2435423, 4870847, 11862575248703, 23725150497407, 281441383062305809756861823, 562882766124611619513723647, 158418504200047111075388369241884118003210485743490303

Offset: 0

N. J. A. Sloane

nonn

Every term is relatively prime to all others.

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).

Vincenzo Librandi, Table of n, a(n) for n = 0..21
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.

Cf. A001685, A003686, A064526.

a[n_?OddQ] := a[n] = 2*a[n-1] + 1; a[n_?EvenQ] := a[n] = (a[n-1]^2 - 3)/2; a[0] = 3; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Jan 25 2013 *)

a(n)=if(n<1,3*(n==0),if(n%2,2*a(n-1)+1,(a(n-1)^2-3)/2))

a(2n+1) = 2*a(2n)+1, a(2n) = (a(2n-1)^2-3)/2, with a(0)=3.

More terms from Jeffrey Shallit

Edited by Michael Somos, Feb 01 2004

A001684 From a continued fraction.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 30, 390, 32370, 81022110, 79098077953830, 2499603048957386233742790, 6399996109983215106481566902449146981585570, 1296147136591533261616288032775924136752630487513536584267056282299509616710

Offset: 0

N. J. A. Sloane

nonn

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).

John Cerkan, Table of n, a(n) for n = 0..17
V. C. Harris, Another proof of the infinitude of primes, Amer. Math. Monthly, 63 (1956), 711.

a(0) = a(1) = a(2) = 1, a(n) = Product_{k=0..n-3} A001685(k) for n >= 3. - Sean A. Irvine, Nov 13 2012

One more term from Sean A. Irvine, Nov 13 2012

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

nonn

Every term is relatively prime to all others. - Michael Somos, Feb 01 2004

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.

Cf. A000215, A001685, A002715, A003686, A064526, A220294.

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 */

a(2*n + 1) = a(2*n) + a(2*n - 1) - 1, a(2*n) = a(2*n - 1)^2 - 3 * a(2*n - 1) + 3, a(0) = 3, a(1) = 5. - Michael Somos, Feb 01 2004
Conjecture: a(2n+1)=A001146(n+1)+1. - R. J. Mathar, May 15 2007
a(2*n) = A220294(n). a(2*n + 1) = A000215(n+1). - Michael Somos, Dec 10 2012

More terms from Jeffrey Shallit

Edited by Michael Somos, Feb 01 2004

A006695 a(2n)=2a(2n-2)^2-1, a(2n+1)=2a(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

An infinite coprime sequence defined by recursion.

Every term is relatively prime to all others. - Michael Somos, Feb 01 2004

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

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

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 *)

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

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

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

Michael Somos, Dec 21 2001

a(i) and a(j) are relative prime for all i>j>0.

An infinite coprime sequence defined by recursion.

Cf. A001685, A002715, A003686, A006695, A064184 (denominators), A064526.

nxt[{a_,b_}]:={b,1+a/b}; NestList[nxt,{0,1},20][[All,1]]//Numerator (* Harvey P. Dale, Sep 26 2016 *)

{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)))}

a(n) = (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)).

a(n) = b(n) + b(n-1) * a(n-2) where b(n) = A064184(n).

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

Amarnath Murthy, Mar 24 2003

An infinite coprime sequence defined by recursion.
Every term is relatively prime to all others. - Michael Somos, Feb 01 2004
Note that gcd(x+y,2*x*y) <= gcd(x+y,2)*gcd(x+y,x)*gcd(x+y,y), so gcd(x,y) = 1 implies gcd(x+y,2*x*y) = 1 unless both x,y are odd. As a result, the definition gives x_{n+1} = x_n+y_n and y_{n+1} = 2*(x_n)*(y_n) with x_0 = 1 and y_0 = 2. - Jianing Song, Oct 10 2021

			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)

Cf. A001685, A003686, A064526.

The denominators are A081476.

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])

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

From Jianing Song, Oct 10 2021: (Start)
a(n) = a(n-1) + A081476(n-1) for n >= 1 with a(0) = 1 and A081476(0) = 2.
a(0) = 1, a(n) = a(n-1) + 2^n*a(0)*a(1)*...*a(n-2) for n >= 1.
a(0) = 1, a(1) = 3, a(n) = a(n-1) + 2*a(n-2)*(a(n-1)-a(n-2)) for n >= 2. (End)

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 06 2003

Edited by Jon E. Schoenfield, Apr 25 2014

A001686 From a continued fraction.

Original entry on oeis.org

1, 1, 2, 3, 8, 51, 1538, 599871, 19417825808, 1573273218577214751, 124442887685693556895657990772138, 311057821480221188367831306672353513246409033360367599771

Offset: 0

N. J. A. Sloane

nonn

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).

V. C. Harris, Another proof of the infinitude of primes, Amer. Math. Monthly, 63 (1956), 711.

Cf. A001684, A001685.

Denominator of continued fraction formed by terms of A001684. - Sean A. Irvine, Nov 13 2012

a(11) from Sean A. Irvine, Nov 13 2012

cp's OEIS Frontend

A003686 Number of genealogical 1-2 rooted trees of height n.

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A064526 Define a pair of sequences by p(0) = 0, q(0) = p(1) = q(1) = 1, q(n+1) = p(n)*q(n-1), p(n+1) = q(n+1) + q(n) for n > 0; then a(n) = p(n) and A064183(n) = q(n).

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A109134 Decimal expansion of Phi, the real root of the equation 1/x = (x-1)^2.

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A002715 An infinite coprime sequence defined by recursion.

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A001684 From a continued fraction.

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A002716 An infinite coprime sequence defined by recursion.

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A006695 a(2n)=2*a(2n-2)^2-1, a(2n+1)=2*a(2n)-1, a(0)=2.

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A006695 a(2n)=2a(2n-2)^2-1, a(2n+1)=2a(2n)-1, a(0)=2.