A001601 -id:A001601 - cp's OEIS frontend

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

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);

A145505 a(n+1)=a(n)^2+2*a(n)-2 and a(1)=5.

Original entry on oeis.org

5, 33, 1153, 1331713, 1773462177793, 3145168096065837266706433, 9892082352510403757550172975146702122837936996353

Offset: 1

Artur Jasinski, Oct 11 2008

nonn

General formula for a(n+1)=a(n)^2+2*a(n)-2 and a(1)=k+1 is a(n)=Floor[((k + Sqrt[k^2 + 4])/2)^(2^((n+1) - 1))

A145502-A145510. A003423, A001601, A190840.

aa = {}; k = 5; Do[AppendTo[aa, k]; k = k^2 + 2 k - 2, {n, 1, 10}]; aa
or
k =4; Table[Floor[((k + Sqrt[k^2 + 4])/2)^(2^(n - 1))], {n, 2, 7}] (*Artur Jasinski*)
NestList[#^2+2#-2&,5,7]  (* Harvey P. Dale, Mar 19 2011 *)

From Peter Bala, Nov 12 2012: (Start)
a(n) = alpha^(2^(n-1)) + (1/alpha)^(2^(n-1)) - 1, where alpha := 3 + 2*sqrt(2).
a(n) = (1 + sqrt(2))^(2^n) + (sqrt(2) - 1)^(2^n) - 1.
a(n) = A003423(n-1) - 1. a(n) = 2*A001601(n) - 1. a(n) = 4*A190840(n-1) + 1.
Recurrence: a(n) = 7*{product {k = 1..n-1} a(k)} - 2 with a(1) = 5.
Product {n = 1..inf} (1 + 1/a(n)) = 7/8*sqrt(2).
Product {n = 1..inf} (1 + 2/(a(n) + 1)) = sqrt(2).
(End)

A007759 Knopfmacher expansion of sqrt(2): a(2n) = 2(a(2n-1) + 1)^2 - 1, a(2n+1) = 2(a(2n)^2 - 1).

Original entry on oeis.org

2, 17, 576, 665857, 886731088896, 1572584048032918633353217, 4946041176255201878775086487573351061418968498176, 48926646634423881954586808839856694558492182258668537145547700898547222910968507268117381704646657

Offset: 1

Arnold Knopfmacher

nonn

G. C. Greubel, Table of n, a(n) for n = 1..11
A. Knopfmacher and J. Knopfmacher, An alternating product representation for real numbers, in Applications of Fibonacci numbers, Vol. 3 (Kluwer 1990), pp. 209-216.

Cf. A002193 (sqrt(2)), A001601.

function a(n)
  if n eq 1 then return 2;
  elif n mod 2 eq 0 then return 2*(a(n-1) +1)^2 -1;
  else return 2*(a(n-1)^2 -1);
  end if; return a; end function;
[a(n): n in [1..9]]; // G. C. Greubel, Mar 04 2020

a:= proc(n) option remember;
if n=1 then 2
elif `mod`(n,2) = 0 then 2*(a(n-1) +1)^2 -1
else 2*(a(n-1)^2 -1)
end if; end proc;
seq(a(n), n = 1..9); # G. C. Greubel, Mar 04 2020

a[n_]:= a[n]= If[n==1, 2, If[EvenQ[n], 2*(a[n-1] +1)^2 -1, 2*a[n-1]^2 -2]]; Table[a[n], {n,9}] (* G. C. Greubel, Mar 04 2020 *)

a(n) = if (n==1, 2, if (n % 2, 2*a(n-1)^2 - 2, 2*(a(n-1)+1)^2 - 1)); \\ Michel Marcus, Feb 20 2019

@CachedFunction
def a(n):
    if (n==1): return 2
    elif (n%2==0): return 2*(a(n-1) +1)^2 -1
    else: return 2*(a(n-1)^2 -1)
[a(n) for n in (1..9)] # G. C. Greubel, Mar 04 2020

More terms from Christian G. Bower, Jan 06 2006

A257553 Primes whose squares are not the sums of two consecutive nonsquares.

Original entry on oeis.org

2, 3, 7, 17, 41, 239, 577, 665857, 9369319, 63018038201, 489133282872437279, 19175002942688032928599, 123426017006182806728593424683999798008235734137469123231828679

Offset: 1

Juri-Stepan Gerasimov, Apr 29 2015

nonn

Primes of the form A257282(k).
2 is in this sequence, and an odd prime p is in the sequence iff either (p^2 - 1)/2 or (p^2 + 1)/2 is a square. - Wolfdieter Lang, May 07 2015
According to the Neretin comment in A257282, and as the primes of A001333 are in A086395, this is (apart from the 2) the same as A086395. - R. J. Mathar, Jan 31 2024

			2 is in the sequence because it is prime and its square 4 is in A256944: 4 is not a sum of consecutive numbers.
3 is in the sequence because it is prime and its square 9 is in A256944: 9 = 2^2 + 5.
7 is in the sequence because it is prime and its square 49 is in A256944: 49 = 24 + 5^2.
5 is not in the sequence because neither 12 nor 13 is a square.

Cf. A000290, A001601, A088165, A256944.

lim = 1000000; s = Plus @@@ (Partition[#, 2, 1] & @ Complement[Range@ lim, Range[Floor@ Sqrt[lim]]^2]); Select[Sqrt[#] & /@ Select[Range@ Floor[Sqrt[lim]]^2, ! MemberQ[s, #] &] , PrimeQ] (* Michael De Vlieger, Apr 29 2015 *)

Name clarified by Michael De Vlieger and Jon E. Schoenfield, May 03 2015

Edited by Wolfdieter Lang, May 07 2015

A152121 a(0) = 4; for n>0, a(n) = a(n-1)^2 - 2^(1+2^(n-1)).

Original entry on oeis.org

4, 12, 136, 18464, 340918784, 116225617283907584, 13508394113025357323362163662782464, 182476711512818130204254420972394401125552102555370860811711166808064

Offset: 0

Dennis Martin (dennis.martin(AT)dptechnology.com), Nov 24 2008

A subset of A056236, where a(n) = (2+sqrt(2))^n+(2-sqrt(2))^n, when the exponent n is a nonnegative integer power of 2. I.E.: a(0) = (2+sqrt(2))^(2^0)+(2-sqrt(2))^(2^0), a(1) = (2+sqrt(2))^(2^1)+(2-sqrt(2))^(2^1); a(2) = (2+sqrt(2))^(2^2)+(2-sqrt(2))^(2^2); etc.

For all n the value 2^(n+1) can be factored from each a(n), which except for a different initial term (a(0) = 2 instead of a(0) = 1) matches the sequence A001601 for n>0.

			a(0) = 4; a(1) = 4^2 - 2^2 = 12; a(2) = 12^2 - 2^3 = 136; a(3) = 136^2 - 2^5 = 18464; a(4) = 18464^2 - 2^9 = 340918784.

Dennis Martin, Table of n, a(n) for n = 0..10

Cf. A056236, A001601

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

A163261 Numerators of fractions in the approximation of the square root of 2 by means of: f(n)= 3f(n-1)/(f(n-1)^2+1); with f(1)= 1.

Original entry on oeis.org

1, 3, 18, 702, 1038258, 2292015792222, 11135478955084481409955218, 263108590829588406010634295716681354685770554450302

Offset: 1

Mark Dols, Jul 23 2009

For root c: f(n) = (1+c)*f(n-1)/(f(n-1)^2+1).

Cf. A002193 (sqrt(2)), A163262 (denominators).

Cf. A001601 and A051009: Newton's method.

f(n) = if (n==1, 1, 3*f(n-1)/(f(n-1)^2+1));
a(n) = numerator(f(n)); \\ Michel Marcus, Mar 05 2019

a(n+1) = 3*a(n)*A163262(n). - Philippe Deléham, Oct 11 2011

More terms from Michel Marcus, Mar 05 2019

A163262 Denominators of fractions in the approximation of the square root of 2 by means of: f(n) = 3*f(n-1)/(f(n-1)^2+1); with f(1)= 1.

Original entry on oeis.org

1, 2, 13, 493, 735853, 1619459312173, 7875984855578888541679213, 186030029004437379749629399827828117533654561726893

Offset: 1

Mark Dols, Jul 23 2009

For root of c: f(n) = (1+c)*f(n-1)/(f(n-1)^2+1).

a(9) has 102 digits. - Emeric Deutsch, Jul 29 2009

Cf. A002193 (sqrt(2)), A163261 (numerators).

Cf. A001601 and A051009: Newton's method.

f[1] := 1: for n from 2 to 10 do f[n] := 3*f[n-1]/(1+f[n-1]^2) end do: seq(denom(f[n]), n = 1 .. 8); # Emeric Deutsch, Jul 29 2009

f(n) = if (n==1, 1, 3*f(n-1)/(f(n-1)^2+1));
a(n) = denominator(f(n)); \\ Michel Marcus, Mar 04 2019

a(7) and a(8) from Emeric Deutsch, Jul 29 2009

Name edited by Michel Marcus, Mar 04 2019

A190840 a(n+1) = 4a(n)(a(n)+1) for a(0) = 1.

Original entry on oeis.org

1, 8, 288, 332928, 443365544448, 786292024016459316676608, 2473020588127600939387543243786675530709484249088

Offset: 0

Alexander Zhukov, Aug 08 2011

nonn

For n>0, subsequence of A132592: both a(n)/2 and a(n)+1 are squares.
All terms (n > 0) are divisible by 8, yielding all terms of A185097, which is indexed from n=1, thus having the first term A185097(1) = 1.
The next term has 98 digits. - Harvey P. Dale, Jan 01 2014
For n>0, subsequence of A060355: both a(n) and a(n)+1 are powerful numbers. - Bernard Schott, Apr 24 2023

Cf. A000217, A001601, A060355, A132592, A185097.

NestList[4#(#+1)&,1,7] (* Harvey P. Dale, Jan 01 2014 *)

a(n+1) = 4*a(n)*(a(n)+1) for a(0) = 1.
a(n) = sinh(2^(n-2)*arccosh(17))^2. - Alexander R. Povolotsky, Aug 14 2011
a(n) = 8*A185097(n) for n > 0. - Alexander R. Povolotsky, Aug 14 2011
a(n) = (1 + sqrt(2))^(2^(n+1))/4 + (1 - sqrt(2))^(2^(n+1))/4 - 1/2. Therefore 2*a(n) + 1 = A001601(n+1). - Bruno Berselli, Feb 01 2017

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Magma

Maple

A145505 a(n+1)=a(n)^2+2*a(n)-2 and a(1)=5.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A007759 Knopfmacher expansion of sqrt(2): a(2n) = 2*(a(2n-1) + 1)^2 - 1, a(2n+1) = 2*(a(2n)^2 - 1).

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Sage

Extensions

A257553 Primes whose squares are not the sums of two consecutive nonsquares.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Extensions

A152121 a(0) = 4; for n>0, a(n) = a(n-1)^2 - 2^(1+2^(n-1)).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A163261 Numerators of fractions in the approximation of the square root of 2 by means of: f(n)= 3f(n-1)/(f(n-1)^2+1); with f(1)= 1.

Views

Author

Keywords

Comments

Crossrefs

Programs

PARI

Formula

Extensions

A163262 Denominators of fractions in the approximation of the square root of 2 by means of: f(n) = 3*f(n-1)/(f(n-1)^2+1); with f(1)= 1.

Views

A007759 Knopfmacher expansion of sqrt(2): a(2n) = 2(a(2n-1) + 1)^2 - 1, a(2n+1) = 2(a(2n)^2 - 1).

A190840 a(n+1) = 4a(n)(a(n)+1) for a(0) = 1.