A081460 -id:A081460 - cp's OEIS frontend

A081459 Consider the mapping f(r) = (1/2)*(r + N/r) from rationals to rationals where N = 5. Starting with r = 2 and applying the mapping to each new (reduced) rational number gives 2, 9/4, 161/72, 51841/23184, ..., tending to N^(1/2). Sequence gives values of the numerators.

2, 9, 161, 51841, 5374978561, 57780789062419261441, 6677239169351578707225356193679818792961, 89171045849445921581733341920411050611581102638589828325078491812417901966295041

Offset: 1

Amarnath Murthy, Mar 22 2003

Related sequence pairs (numerator, denominator) can be obtained by choosing N = 2, 3, 6 etc.

The sequence satisfies the Pell equation a(n+1)^2 - 5*A081460(n+1)^2 = 1. - Vincenzo Librandi, Dec 20 2011

Vincenzo Librandi, Table of n, a(n) for n = 1..11

Cf. A000032, A000129, A001333, A081460.

m:=8; f:=[ n eq 1 select 2 else (Self(n-1)+5/Self(n-1))/2: n in [1..m] ]; [ Numerator(f[n]): n in [1..m] ]; // Bruno Berselli, Dec 20 2011

k = 4; Table[Simplify[Expand[(1/2) (((k + Sqrt[k^2 + 4])/2)^(2^(n - 1)) + ((k - Sqrt[k^2 + 4])/2)^(2^(n - 1)))]], {n, 1, 6}] (* Artur Jasinski, Oct 12 2008 *)
aa = {}; k = 9; Do[AppendTo[aa, k]; k = 2 k^2 - 1, {n, 1, 5}]; aa (* Artur Jasinski, Oct 12 2008 *)

{r=2; N=5; for(n=1,8,a=numerator(r); b=denominator(r); print1(a,","); r=(1/2)*(r + N/r) )}

a(n) = a(n-1)^2 + 5*A081460(n-1)^2. - Mario Catalani (mario.catalani(AT)unito.it), May 21 2003
a(n) = (1/2)*(((4+2*sqrt(5))/2)^(2^(n-1)) + ((4-2*sqrt(5))/2)^(2^(n-1))). a(n+1) = 2*a(n)^2 - 1 for n > 1. - Artur Jasinski, Oct 12 2008
a(n) = A000032(3*2^(n-1))/2. - Ehren Metcalfe, Oct 05 2017
a(n) = A001077(2^(n-1)). - Robert FERREOL, Apr 16 2023
From  Peter Bala, Jun 22 2025: (Start)
Product_{n >= 1} (1 + 1/a(n)) = (3/4)*sqrt(5).
Product_{n >= 1} (1 - 1/(2*a(n))) = (6/19)*sqrt(5). See A002812. (End)

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

A244012 Numerators of rational approximations to sqrt(7) obtained from Newton's method.

Original entry on oeis.org

2, 11, 233, 108497, 23543191457, 1108563727961872518977, 2457827077905448997994482872789298261401217, 12081827889770476116093110581355561229584727594431650162181251776430351279198649072897

Offset: 0

N. J. A. Sloane, Jun 18 2014

			2, 11/4, 233/88, 108497/41008, 23543191457/8898489952, ...

R. Parimala, A Hasse principle for quadratic forms over function fields, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 3, 447--461. MR3196794.

Cf. A244013 (denominators).

The analogs for sqrt(k), k=2,3,5,6,7 are: A001601/A051009, A002812/A071579, A081459/A081460, A244014/A244015, A244012/A244013.

N:=7;
s:=[floor(sqrt(N))];
M:=8;
for n from 1 to M do
x:=s[n];
h:=(N-x^2)/(2*x);
s:=[op(s),x+h]; od:
lprint(s);
s1:=map(numer,s);
s2:=map(denom,s);

A244013 Denominators of rational approximations to sqrt(7) obtained from Newton's method.

Original entry on oeis.org

1, 4, 88, 41008, 8898489952, 418997705236253480128, 928971316248341903257187589777603944778112, 4566501711345281867283814391125123371716411674583075407993026856131137508750543524608

Offset: 0

N. J. A. Sloane, Jun 18 2014

			2, 11/4, 233/88, 108497/41008, 23543191457/8898489952, ...

R. Parimala, A Hasse principle for quadratic forms over function fields, Bull. Amer. Math. Soc. (N.S.) 51 (2014), no. 3, 447--461. MR3196794.

Cf. A244012 (numerators).

The analogs for sqrt(k), k=2,3,5,6,7 are: A001601/A051009, A002812/A071579, A081459/A081460, A244014/A244015, A244012/A244013.

N:=7;
s:=[floor(sqrt(N))];
M:=8;
for n from 1 to M do
x:=s[n];
h:=(N-x^2)/(2*x);
s:=[op(s),x+h]; od:
lprint(s);
s1:=map(numer,s);
s2:=map(denom,s);

A244014 Numerators of rational approximations to sqrt(6) obtained from Newton's method.

Original entry on oeis.org

2, 5, 49, 4801, 46099201, 4250272665676801, 36129635465198759610694779187201, 2610701117696295981568349760414651575095962187244375364404428801

Offset: 0

N. J. A. Sloane, Jun 18 2014

			2, 5/2, 49/20, 4801/1960, 46099201/18819920, ...

Cf. A244015, A084765.

The analogs for sqrt(k), k=2,3,5,6,7 are: A001601/A051009, A002812/A071579, A081459/A081460, A244014/A244015, A244012/A244013.

N:=6;
s:=[floor(sqrt(N))];
M:=8;
for n from 1 to M do
x:=s[n];
h:=(N-x^2)/(2*x);
s:=[op(s),x+h]; od:
lprint(s);
s1:=map(numer,s);
s2:=map(denom,s);

A244015 Denominators of rational approximations to sqrt(6) obtained from Newton's method.

Original entry on oeis.org

1, 2, 20, 1960, 18819920, 1735166549767840, 14749861913749949808286047759680, 1065814268211609269094400465471990022332221793124358274759711360

Offset: 0

N. J. A. Sloane, Jun 18 2014

			2, 5/2, 49/20, 4801/1960, 46099201/18819920, ...

Cf. A244014 (numerators).

The analogs for sqrt(k), k=2,3,5,6,7 are: A001601/A051009, A002812/A071579, A081459/A081460, A244014/A244015, A244012/A244013.

m:=9; f:=[n eq 1 select 2 else (Self(n-1)+6/Self(n-1))/2: n in [1..m]]; [Denominator(f[n]): n in [1..m]]; // Vincenzo Librandi, Jan 12 2016

N:=6;
s:=[floor(sqrt(N))];
M:=8;
for n from 1 to M do
x:=s[n];
h:=(N-x^2)/(2*x);
s:=[op(s),x+h]; od:
lprint(s);
s1:=map(numer,s);
s2:=map(denom,s);

A083697 a(n) = 2^(2^n - 1) * Fibonacci(2^n).

Original entry on oeis.org

1, 2, 24, 2688, 32342016, 4677882957791232, 97861912906883207538212742365184, 42829440312913272520181533609472356498655100482256687829780267008

Offset: 0

Mario Catalani (mario.catalani(AT)unito.it), May 22 2003

A083696(n)/a(n) converges to sqrt(5).

Similar to A081460: a(n) is the denominator of the same mapping f(r)=(1/2)(r+5/r) but with initial value r=1.

G. C. Greubel, Table of n, a(n) for n = 0..10

Cf. A000045, A058635, A058891, A081460, A083696.

[2^(2^n -1)*Fibonacci(2^n): n in [0..8]]; // G. C. Greubel, Jan 14 2022

Table[Sum[Product[2^n -k, {k,0,2*r}]k^r/(2*r+1)!, {r,0,2^n -1}], {n,0,8}]
Table[2^(2^n -1)*Fibonacci[2^n], {n,0,8}] (* G. C. Greubel, Jan 14 2022 *)

[2^(2^n -1)*lucas_number1(2^n, 1, -1) for n in (0..8)] # G. C. Greubel, Jan 14 2022

a(n) = 2*a(n-1)*A083696(n-1).
a(n) = A058635(n) * A058891(n).
a(n) = 2^(2^n - 1) * A000045(2^n).
a(n) = Sum_{r=0..(2^n -1)} (5^r/(2*r+1)!)*Product_{k=0..2*r} (2^n - k).

The next term is too large to include.

Better description from Ralf Stephan, Aug 29 2004

A319749 a(n) is the numerator of the Heron sequence with h(0)=3.

Original entry on oeis.org

3, 11, 119, 14159, 200477279, 40191139395243839, 1615327685887921300502934267457919, 2609283532796026943395592527806764363779539144932833602430435810559

Offset: 0

Paul Weisenhorn, Sep 27 2018

The denominator of the Heron sequence is in A319750.
The following relationship holds between the numerator of the Heron sequence and the numerator of the continued fraction A041018(n)/A041019(n) convergent to sqrt(13).
n even: a(n)=A041018((5*2^n-5)/3).
n  odd: a(n)=A041018((5*2^n-1)/3).
More generally, all numbers c(n)=A078370(n)=(2n+1)^2+4 have the same relationship between the numerator of the Heron sequence and the numerator of the continued fraction convergent to 2n+1.
sqrt(c(n)) has the continued fraction 2n+1; n,1,1,n,4n+2.
hn(n)^2-c(n)*hd(n)^2=4 for n>1.
From Peter Bala, Mar 29 2022: (Start)
Applying Heron's method (sometimes called the Babylonian method) to approximate the square root of the function x^2 + 4, starting with a guess equal to x, produces the sequence of rational functions [x, 2*T(1,(x^2+2)/2)/x, 2*T(2,(x^2+2)/2)/( 2*x*T(1,(x^2+2)/2) ), 2*T(4,(x^2+2)/2)/( 4*x*T(1,(x^2+2)/2)*T(2,(x^2+2)/2) ), 2*T(8,(x^2+2)/2)/( 8*x*T(1,(x^2+2)/2)*T(2,(x^2+2)/2)*T(4,(x^2+2)/2) ), ...], where T(n,x) denotes the n-th Chebyshev polynomial of the first kind. The present sequence is the case x = 3. Cf. A001566 and A058635 (case x = 1), A081459 and A081460 (essentially the case x = 4). (End)

			A078370(2)=29.
hn(0)=A041046(0)=5; hn(1)=A041046(3)=27; hn(2)=A041046(5)=727;
hn(3)=A041046(13)=528527.

P. Liardet and P. Stambul, Séries d'Engel et fractions continuées, Journal de Théorie des Nombres de Bordeaux 12 (2000), 37-68.
Wikipedia, Engel Expansion
Wikipedia, Methods of computing square roots
Index entries for sequences of form a(n+1)=a(n)^2 + ...

2*T(2^n,x/2) modulo differences of offset: A001566 (x = 3 and x = 7), A003010 (x = 4), A003487 (x = 5), A003423 (x = 6), A346625 (x = 8), A135927 (x = 10), A228933 (x = 18).

Cf. A041018, A041019, A057076, A078370, A081459, A010122, A319750, A041046.

hn[0]:=3:  hd[0]:=1:
for n from 1 to 6 do
hn[n]:=(hn[n-1]^2+13*hd[n-1]^2)/2:
hd[n]:=hn[n-1]*hd[n-1]:
   printf("%5d%40d%40d\n", n, hn[n], hd[n]):
end do:
#alternative program
a := n -> if n = 0 then 3 else simplify( 2*ChebyshevT(2^(n-1), 11/2) ) end if:
seq(a(n), n = 0..7); # Peter Bala, Mar 16 2022

def aupton(nn):
    hn, hd, alst = 3, 1, [3]
    for n in range(nn):
        hn, hd = (hn**2 + 13*hd**2)//2, hn*hd
        alst.append(hn)
    return alst
print(aupton(7)) # Michael S. Branicky, Mar 16 2022

h(n) = hn(n)/hd(n); hn(0)=3; hd(0)=1.
hn(n+1) = (hn(n)^2+13*hd(n)^2)/2.
hd(n+1) = hn(n)*hd(n).
A041018(n) = A010122(n)*A041018(n-1) + A041018(n-2).
A041019(n) = A010122(n)*A041019(n-1) + A041019(n-2).
From Peter Bala, Mar 16 2022: (Start)
a(n) = 2*T(2^(n-1),11/2) for n >= 1, where T(n,x) denotes the n-th Chebyshev polynomial of the first kind.
a(n) = 2*T(2^n, 3*sqrt(-1)/2) for n >= 2.
a(n) = ((11 + 3*sqrt(13))/2)^(2^(n-1)) + ((11 - 3*sqrt(13))/2)^(2^(n-1)) for n >= 1.
a(n+1) = a(n)^2 - 2 for n >= 1.
a(n) = A057076(2^(n-1)) for n >= 1.
Engel expansion of (1/6)*(13 - 3*sqrt(13)); that is, (1/6)*(13 - 3*sqrt(13)) = 1/3 + 1/(3*11) + 1/(3*11*119) + .... (Define L(n) = (1/2)*(n - sqrt(n^2 - 4)) for n >= 2 and show L(n) = 1/n + L(n^2-2)/n. Iterate this relation with n = 11. See also Liardet and Stambul, Section 4.)
sqrt(13) = 6*Product_{n >= 0} (1 - 1/a(n)).
sqrt(13) = (9/5)*Product_{n >= 0} (1 + 2/a(n)). See A001566. (End)

a(6) and a(7) added by Peter Bala, Mar 16 2022

cp's OEIS Frontend

A081459 Consider the mapping f(r) = (1/2)*(r + N/r) from rationals to rationals where N = 5. Starting with r = 2 and applying the mapping to each new (reduced) rational number gives 2, 9/4, 161/72, 51841/23184, ..., tending to N^(1/2). Sequence gives values of the numerators.

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A244012 Numerators of rational approximations to sqrt(7) obtained from Newton's method.

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A244013 Denominators of rational approximations to sqrt(7) obtained from Newton's method.

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A244014 Numerators of rational approximations to sqrt(6) obtained from Newton's method.

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A244015 Denominators of rational approximations to sqrt(6) obtained from Newton's method.

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Magma

Maple

A083697 a(n) = 2^(2^n - 1) * Fibonacci(2^n).

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A319749 a(n) is the numerator of the Heron sequence with h(0)=3.

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