A003686 -id:A003686 - cp's OEIS frontend

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

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

A296516 a(n) is the number of terms in expanded form of bivariate polynomial Q_n, where (P_n, Q_n) is defined by: P_0 = x, Q_0 = y, P_(n+1) = P_n + Q_n, Q_(n+1) = P_n * Q_n.

Original entry on oeis.org

1, 1, 2, 5, 14, 40, 111, 300, 797, 2098, 5499, 14389, 37634, 98435, 257516, 673827, 1763460, 4615686, 12082137, 31628294

Offset: 0

Luc Rousseau, Feb 27 2018

Programs based on the direct application of the definition quickly reach a limitation by combinatorial explosion, hence this short list of values in section Data. The first conjectured formula (see Formulas) obtained by the observation of a pattern in the 2D shape of Q_n (as drawn in the Examples) is more computationally efficient and makes it possible to produce a significantly longer list of (nonguaranteed) values: see attached a-file in b-file format, section Links.

A003686 and A064847 are values of P_n and Q_n at x=y=1 (i.e., sums of coefficients in these polynomials). At x=2, y=1 (or vice versa) P_n and Q_n seem to give same sequences but shifted. At x=y=-1, P_n seems to give A000058 negated interleaved with -1's, while Q_n seems to give A007018 interleaved with the same sequence negated. - Andrey Zabolotskiy, May 22 2018

			Q_0 = y -> one term -> a(0) = 1;
Q_1 = x*y -> one term -> a(1) = 1;
Q_2 = x^2*y + x*y^2 -> two terms -> a(2) = 2;
Q_3 = x^3*y + 2*x^2*y^2 + x^3*y^2 + x*y^3 + x^2*y^3 -> five terms -> a(3) = 5;
...
Locations of terms in 2D arrays indexed by the exponents of x and y:
   0: .   1: ..   2: ...   3: ....   4: ......   5: .........
      X      .X      ..X      ...X      ....X.      .....X...
                     .X.      ..XX      ...XXX      ....XXXX.
                              .XX.      ..XXXX      ...XXXXXX
                                        .XXXX.      ..XXXXXXX
                                        ..XX..      .XXXXXXXX
                                                    ..XXXXXX.
                                                    ..XXXXX..
                                                    ...XXX...

Luc Rousseau, 201 terms obtained with conjectured formula.
Rémy Sigrist, Colored scatterplot of the points (i, j) such that the coefficient of x^i*y^j in Q_14 is nonzero (where the color is function of the coefficient of x^i*y^j in Q_14).

Cf. A000045, A000217, A001622, A033192, A003686, A064847, A000058, A007018.

Cf. A005207 (which with +1 added appears to be to P_n as a(n) is to Q_n).

{p[0], q[0]} := {x, y};
{p[n_], q[n_]} := {p[n - 1] + q[n - 1], p[n - 1] q[n - 1]};
a[n_] := Length@CoefficientRules[q[n]];
Table[a[n], {n, 0, 10}] (* Andrey Zabolotskiy, Peter Luschny, May 30 2018 *)
From Luc Rousseau, Feb 27 2018: (Start)
(* conjectured: *)
T[n_] := n*(n + 1)/2
F[n_] := Fibonacci[n]
V[n_] := Sum[F[k]*(Sum[(n - 2 - l)*F[l], {l, k + 1, n - 3}]), {k, 1, n - 4}]
W[n_] := Sum[(n - 2 - l)*T[F[l]], {l, 1, n - 3}]
AA[n_] := (F[n + 1])^2 - T[n - 1] - T[F[n - 1]] - 2*V[n] - 2 W[n]
Table[AA[n], {n, 1, 50}]
(End)

def A296516(n):
    P, Q = {(1,0)}, {(0,1)}
    for _ in range(n): P, Q = P|Q, set((p[0]+q[0],p[1]+q[1]) for p in P for q in Q)
    return len(Q) # Chai Wah Wu, Oct 18 2021

Conjectured:
a(n) = F(n+1)^2 - T(n-1) - T(F(n-1)) - 2*V(n) - 2*W(n) for n > 0, where
  F(n) is the n-th Fibonacci number,
  T(n) is the n-th triangular number,
  V(n) = Sum_{i=1..n-4} F(i)*(Sum_{j=i+1..n-3} (n-2-j)*F(j)),
  W(n) = Sum_{i=1..n-3} (n-2-i)*T(F(i)).
Or:
a(n) = F(n+1)^2 - T(n-1) - T(F(n-1)) - [n>=4]*U(n) where [] is the Iverson bracket, and U(n) = F(n-1)*F(n-3) + F(n+1)*(2*F(n-4)-1) + 5 - 8*(n mod 2).
Conjectures from Colin Barker, Mar 28 2018: (Start)
G.f.: 1 + x*(1 - 5*x + 9*x^2 - 5*x^3 - 2*x^4 + x^5) / ((1 - x)^3*(1 - 3*x + x^2)*(1 - x - x^2)).
a(n) = 7*a(n-1) - 18*a(n-2) + 20*a(n-3) - 6*a(n-4) - 6*a(n-5) + 5*a(n-6) - a(n-7) for n > 7.
(End)
From Luc Rousseau, Mar 30 2018: (Start)
Derived from the conjectured order-7 linear recurrence above, for n > 0,
a(n) = -(1/2)*n^2 + (1/2)*n - 1 + ((2+phi)/10)*(phi^2)^n + ((4-3*phi)/10)*(-phi^(-1))^n + ((1+3*phi)/10)*phi^n + ((3-phi)/10)*(phi^(-2))^n,
where phi denotes the golden ratio and lim_{n->oo} a(n+1)/a(n) = phi^2.
a(n) = (F(2*n+1) + F(n+2) - n^2 + n - 2) / 2.
(End)

a(15)-a(19) from Andrey Zabolotskiy, May 30 2018

cp's OEIS Frontend

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

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A066356 Numerator of sequence defined by recursion c(n) = 1 + c(n-2) / c(n-1), c(0) = 0, c(1) = 1.

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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.

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Author

Keywords

Comments

Examples

Crossrefs

Programs

PARI

PARI

Formula

Extensions

A296516 a(n) is the number of terms in expanded form of bivariate polynomial Q_n, where (P_n, Q_n) is defined by: P_0 = x, Q_0 = y, P_(n+1) = P_n + Q_n, Q_(n+1) = P_n * Q_n.

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Examples

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Programs

Mathematica

Python

Formula

Extensions

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