A041008 -id:A041008 - cp's OEIS frontend

A010465 Decimal expansion of square root of 7.

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

Offset: 1

N. J. A. Sloane

Continued fraction expansion is 2 followed by {1, 1, 1, 4} repeated. - Harry J. Smith, Jun 01 2009

The convergents to sqrt(7) are given in A041008/A041009. - Wolfdieter Lang, Nov 22 2017

			2.645751311064590590501615753639260425710259183082450180368334459201...

Harry J. Smith, Table of n, a(n) for n = 1..20000
Jason Kimberley, Index of expansions of sqrt(d) in base b.
R. J. Nemiroff & J. Bonnell, The first 1 million digits of the square root of 7.
Simon Plouffe, Plouffe's Inverter, sqrt(7) to 100000 digits.
Index entries for algebraic numbers, degree 2.

Cf. A010121 (continued fraction), A041008/A041009.

SetDefaultRealField(RealField(100)); Sqrt(7); // Vincenzo Librandi, Feb 15 2020

RealDigits[N[Sqrt[7], 200]][[1]] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2011 *)

default(realprecision, 20080); x=sqrt(7); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b010465.txt", n, " ", d));  \\ Harry J. Smith, Jun 01 2009

Equals 8*cos(Pi/14)*sin(2*Pi/14)*cos(3*Pi/14). - Gerry Martens, Mar 13 2025

A041006 Numerators of continued fraction convergents to sqrt(6).

Original entry on oeis.org

2, 5, 22, 49, 218, 485, 2158, 4801, 21362, 47525, 211462, 470449, 2093258, 4656965, 20721118, 46099201, 205117922, 456335045, 2030458102, 4517251249, 20099463098, 44716177445, 198964172878, 442644523201, 1969542265682, 4381729054565, 19496458483942

Offset: 0

N. J. A. Sloane

Interspersion of 2 sequences, 2*A054320 and A001079. - Gerry Martens, Jun 10 2015

Hugo Pfoertner, Table of n, a(n) for n = 0..100
Index entries for linear recurrences with constant coefficients, signature (0,10,0,-1).

Cf. A041007 (denominators).
Cf. A054320, A001079.
Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041008 (m=7), A041010 (m=8), A005667 (m=10), A041014 (m=11), ..., A042936 (m=1000).

I:=[2, 5, 22, 49]; [n le 4 select I[n] else 10*Self(n-2)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 10 2015

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[6],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)
LinearRecurrence[{0, 10, 0, -1}, {2, 5, 22, 49}, 50] (* Vincenzo Librandi, Jun 10 2015 *)

A41006=contfracpnqn(c=contfrac(sqrt(6)), #c)[1, ][^-1] \\ Discard possibly incorrect last element. NB: a(n)=A41006[n+1]! M. F. Hasler, Nov 01 2019

\\ For correct index & more terms:
A041006(n)={n<#A041006|| A041006=extend(A041006, [2, 10; 4, -1], n\.8); A041006[n+1]}
extend(A, c, N)={for(n=#A+1, #A=Vec(A, N), A[n]=[A[n-i]|i<-c[, 1]]*c[, 2]); A} \\ M. F. Hasler, Nov 01 2019

From M. F. Hasler, Feb 13 2009: (Start)
a(2n) = 2*A142238(2n) = A041038(2n)/2;
a(2n-1) = A142238(2n-1) = A041038(2n-1) = A001079(n). (End)
G.f.: (2 + 5*x + 2*x^2 - x^3)/(1 - 10*x^2 + x^4).
a(n) = ((2 + sqrt(6))^(n+1) + (2 - sqrt(6))^(n+1))/2^(ceiling(n/2) + 1). - Robert FERREOL, Oct 13 2024
E.g.f.: sqrt(2)*sinh(sqrt(2)*x)*(cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)) + cosh(sqrt(2)*x)*(2*cosh(sqrt(3)*x) + sqrt(3)*sinh(sqrt(3)*x)). - Stefano Spezia, Oct 14 2024

More terms from Vincenzo Librandi, Jun 10 2015

A041014 Numerators of continued fraction convergents to sqrt(11).

Original entry on oeis.org

3, 10, 63, 199, 1257, 3970, 25077, 79201, 500283, 1580050, 9980583, 31521799, 199111377, 628855930, 3972246957, 12545596801, 79245827763, 250283080090, 1580944308303, 4993116004999, 31539640338297

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0,20,0,-1).

Cf. A010468, A041015 (denominators).

Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041008 (m=7), A041010 (m=8), A005667 (m=10), A041016 (m=12), ..., A042936 (m=1000).

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[11],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011 *)
Numerator[Convergents[Sqrt[11], 30]] (* Vincenzo Librandi, Oct 28 2013 *)

A041014=contfracpnqn(c=contfrac(sqrt(11)), #c)[1,][^-1] \\ Discard last element which may be incorrect. Use e.g. \p999 to get more terms, or extend as follows:
{A041014_upto(N,A=Vec(A041014,N))=for(n=#A041014+1,N, A[n]=20*A[n-2]-A[n-4]); A041014=A} \\ M. F. Hasler, Nov 01 2019

G.f.: (3 + 10*x + 3*x^2 - x^3)/(1 - 20*x^2 + x^4).

A042936 Numerators of continued fraction convergents to sqrt(1000).

Original entry on oeis.org

31, 32, 63, 95, 158, 253, 1676, 3605, 8886, 136895, 282676, 702247, 4496158, 5198405, 9694563, 14892968, 24587531, 39480499, 2472378469, 2511858968, 4984237437, 7496096405, 12480333842, 19976430247, 132338915324, 284654260895, 701647437114, 10809365817605, 22320379072324

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 78960998, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).

Cf. A042937 (denominators).

Analog for sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041008 (m=7), A041010 (m=8), A005667 (m=10), A041014 (m=11), ..., A042934 (m=999).

Numerator[Convergents[Sqrt[1000], 30]] (* Harvey P. Dale, Oct 29 2013 *)

A42936=contfracpnqn(c=contfrac(sqrt(1000)), #c)[1,][^-1] \\ Discards possibly incorrect last term. NB: a(n)=A42936[n+1]. Could be extended using: {A42936=concat(A42936, 78960998*A42936[-18..-1]-A42936[-36..-19])}
\\ But terms with arbitrarily large indices can be computed using:
A042936(n)={[A42936[n%18+i]|i<-[1, 19]]*([0, -1; 1, 78960998]^(n\18))[,1]} \\ Faster but longer with n=divrem(n,18). (End)

A041009 Denominators of continued fraction convergents to sqrt(7).

Original entry on oeis.org

1, 1, 2, 3, 14, 17, 31, 48, 223, 271, 494, 765, 3554, 4319, 7873, 12192, 56641, 68833, 125474, 194307, 902702, 1097009, 1999711, 3096720, 14386591, 17483311, 31869902, 49353213, 229282754, 278635967

Offset: 0

N. J. A. Sloane

Sqrt(7) = 2 + 9/14 + 9/(14*223) + 9/(223*3554) + 9/(3554*56641) + ...; sum of these 5 terms = 2.64575131088, with sqrt(7) = 2.64575131106... The terms 14, 223, 3554, ... = a(4), a(8), a(12), ... - Gary W. Adamson, Dec 27 2007

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
C. Elsner, M. Stein, On Error Sum Functions Formed by Convergents of Real Numbers, J. Int. Seq. 14 (2011) # 11.8.6
Index entries for linear recurrences with constant coefficients, signature (0,0,0,16,0,0,0,-1).

Cf. A010465, A041008.

Table[Denominator[FromContinuedFraction[ContinuedFraction[Sqrt[7],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011*)
Denominator[Convergents[Sqrt[7],30]] (* or *) LinearRecurrence[ {0,0,0,16,0,0,0,-1},{1,1,2,3,14,17,31,48},30] (* Harvey P. Dale, Dec 17 2019 *)

G.f.: (1+x+2*x^2+3*x^3-2*x^4+x^5-x^6)/(1-16*x^4+x^8). - Colin Barker, Mar 13 2012

A041010 Numerators of continued fraction convergents to sqrt(8).

Original entry on oeis.org

2, 3, 14, 17, 82, 99, 478, 577, 2786, 3363, 16238, 19601, 94642, 114243, 551614, 665857, 3215042, 3880899, 18738638, 22619537, 109216786, 131836323, 636562078, 768398401, 3710155682, 4478554083, 21624372014, 26102926097, 126036076402, 152139002499

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..199 [1 removed by _Georg Fischer_, Jul 01 2019]
Index entries for linear recurrences with constant coefficients, signature (0,6,0,-1).

Cf. A040005 (continued fraction), A041011 (denominators), A010466 (decimals).

Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041008 (m=7), A005667 (m=10), A041014 (m=11), A041016 (m=12), ..., A042934 (m=999), A042936 (m=1000).

Table[Numerator[FromContinuedFraction[ContinuedFraction[Sqrt[8],n]]],{n,1,50}] (* Vladimir Joseph Stephan Orlovsky, Mar 16 2011*)
CoefficientList[Series[(2 + 3*x + 2*x^2 - x^3)/(1 - 6*x^2 + x^4), {x, 0, 30}], x]  (* Vincenzo Librandi, Oct 28 2013 *)
a0[n_] := -((3-2*Sqrt[2])^n*(1+Sqrt[2]))+(-1+Sqrt[2])*(3+2*Sqrt[2])^n // Simplify
a1[n_] := ((3-2*Sqrt[2])^n+(3+2*Sqrt[2])^n)/2 // Simplify
Flatten[MapIndexed[{a0[#], a1[#]} &,Range[20]]] (* Gerry Martens, Jul 11 2015 *)

A041010=contfracpnqn(c=contfrac(sqrt(8)),#c)[1,][^-1] \\ Discard possibly incorrect last element. NB: a(n)=A041010[n+1]! For more terms use:
A041010(n)={n<#A041010|| A041010=extend(A041010, [-1,0,6,0]~, n\.8); A041010[n+1]}
extend(A,c,N)={for(n=#A+1,#A=Vec(A,N), A[n]=A[n-#c..n-1]*c);A} \\ (End)

a(n) = 6*a(n-2) - a(n-4).
a(2n) = a(2n-1) + a(2n-2), a(2n+1) = 4*a(2n) + a(2n-1).
a(2n) = A001333(2n), a(2n+1) = 2*A001333(2n+1).
G.f.: (2+3*x+2*x^2-x^3)/(1-6*x^2+x^4).
a(n) = A001333(n+1)*A000034(n+1). - R. J. Mathar, Jul 08 2009
From Gerry Martens, Jul 11 2015: (Start)
Interspersion of 2 sequences [a0(n),a1(n)] for n>0:
a0(n) = -((3-2*sqrt(2))^n*(1+sqrt(2))) + (-1+sqrt(2))*(3+2*sqrt(2))^n.
a1(n) = ((3-2*sqrt(2))^n + (3+2*sqrt(2))^n)/2. (End)

Entry improved by Michael Somos
Initial term 1 removed and b-file, program and formulas adapted by Georg Fischer, Jul 01 2019
Cross-references added by M. F. Hasler, Nov 02 2019

A042934 Numerators of continued fraction convergents to sqrt(999).

Original entry on oeis.org

31, 32, 63, 95, 158, 885, 5468, 6353, 37233, 80819, 441328, 522147, 3574210, 18393197, 21967407, 40360604, 62328011, 102688615, 6429022141, 6531710756, 12960732897, 19492443653, 32453176550

Offset: 0

N. J. A. Sloane

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 205377230, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1).

Cf. A042935 (denominators).

Analog for other sqrt(m): A001333 (m=2), A002531 (m=3), A001077 (m=5), A041006 (m=6), A041008 (m=7), A041010 (m=8), A005667 (m=10), A041014 (m=11), ..., A042936 (m=1000).

Numerator[Convergents[Sqrt[999], 30]] (* Vincenzo Librandi, Dec 10 2013 *)

A42934=contfracpnqn(c=contfrac(sqrt(999)), #c)[1,][^-1] \\ Discard possibly incorrect last element. NB: a(n) = A42934[n+1]! For more terms, use:
A042934(n)={n<#A42934 || A42934_upto(n+10); A42934[n+1]}
{A42934_upto(N,A=Vec(A42934,N))=for(n=#A42934+1,N, A[n]=205377230*A[n-18]-A[n-36]); A42934=A} \\ M. F. Hasler, Nov 01 2019

a(n) = 205377230*a(n-18) - a(n-36). - Wesley Ivan Hurt, May 28 2021

A266698 x-values of solutions to the Diophantine equation x^2 - 7*y^2 = 2.

Original entry on oeis.org

3, 45, 717, 11427, 182115, 2902413, 46256493, 737201475, 11748967107, 187246272237, 2984191388685, 47559815946723, 757972863758883, 12080006004195405, 192522123203367597, 3068273965249686147, 48899861320791610755, 779329507167416085933, 12420372253357865764173, 197946626546558436140835

Offset: 1

Sture Sjöstedt, Jan 03 2016

A159678 gives the y-values of solutions to the Diophantine equation x^2 - 7*y^2 = 2.

G. C. Greubel, Table of n, a(n) for n = 1..750
Index entries for linear recurrences with constant coefficients, signature (16,-1).

Cf. A041008, A077412, A157456, A159678.

[n: n in [1..2*10^7] | IsSquare((n^2-2)/7)]; // Vincenzo Librandi, Jan 06 2016

Mathematica
```
LinearRecurrence[{16,-1}, {3, 45}, 20 ]
```

lista(nn) = {print1(x = 3, ", "); print1(y = 45, ", "); for (n=2, nn, z = 16*y - x; print1(z, ", "); x = y; y = z;);} \\ Michel Marcus, Jan 05 2016

[3*(chebyshev_U(n-1, 8) - chebyshev_U(n-2, 8)) for n in (1..30)] # G. C. Greubel, Jun 25 2022

a(1)=3, a(2)=45, a(n) = 16*a(n-1) - a(n-2).
a(n) = A041008(4n-2). - Robert Israel, Jan 05 2016
From R. J. Mathar, Jan 12 2016: (Start)
G.f.: 3*x*(1-x) / ( 1-16*x+x^2 ).
a(n) = 3*A157456(n). (End)
From G. C. Greubel, Jun 25 2022: (Start)
a(n) = 3*(ChebyshevU(n-1, 8) - ChebyshevU(n-2, 8)).
E.g.f.: exp(8*x)*(3*cosh(3*sqrt(7)*x) - sqrt(7)*sinh(3*sqrt(7)*x)) - 3. (End)

A259596 Denominators of the other-side convergents to sqrt(7).

Original entry on oeis.org

1, 2, 3, 5, 17, 31, 48, 79, 271, 494, 765, 1259, 4319, 7873, 12192, 20065, 68833, 125474, 194307, 319781, 1097009, 1999711, 3096720, 5096431, 17483311, 31869902, 49353213, 81223115, 278635967, 507918721, 786554688, 1294473409, 4440692161, 8094829634

Offset: 0

Clark Kimberling, Jul 20 2015

Suppose that a positive irrational number r has continued fraction [a(0), a(1), ... ]. Define sequences p(i), q(i), P(i), Q(i) from the numerators and denominators of finite continued fractions as follows:
p(i)/q(i) = [a(0), a(1), ... a(i)] and P(i)/Q(i) = [a(0), a(1), ..., a(i) + 1]. The fractions p(i)/q(i) are the convergents to r, and the fractions P(i)/Q(i) are introduced here as the "other-side convergents" to
r, because p(2k)/q(2k) < r < P(2k)/Q(2k) and P(2k+1)/Q(2k+1) < r < p(2k+1)/q(2k+1), for k >= 0. The closeness of P(i)/Q(i) to r is indicated by |r - P(i)/Q(i)| < |p(i)/q(i) - P(i)/Q(i)| = 1/(q(i)Q(i)), for i >= 0.

			For r = sqrt(7), 3, 5/2, 8/3, 13/5, 45/17, 82/31, 127/48. A comparison of convergents with other-side convergents:
i    p(i)/q(i)         P(i)/Q(i)    p(i)*Q(i)-P(i)*q(i)
0    2/1   < sqrt(7) <    3/1               -1
1    3/1   > sqrt(7) >    5/2                1
2    5/2   < sqrt(7) <    8/3               -1
3    8/3   > sqrt(7) >   13/5                1
4    37/14 < sqrt(7) <   45/17              -1
5    45/17 > sqrt(7) >   83/31               1

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,16,0,0,0,-1).

Cf. A041008, A041009, A259597 (numerators).

r = Sqrt[7]; a[i_] := Take[ContinuedFraction[r, 35], i];
b[i_] := ReplacePart[a[i], i -> Last[a[i]] + 1];
t = Table[FromContinuedFraction[b[i]], {i, 1, 35}]
u = Denominator[t]
LinearRecurrence[{0,0,0,16,0,0,0,-1},{1,2,3,5,17,31,48,79},40] (* Harvey P. Dale, Jun 03 2017 *)

Vec(-(x+1)*(x^2-x-1)*(x^4+3*x^2+1)/(x^8-16*x^4+1) + O(x^50)) \\ Colin Barker, Jul 21 2015

p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = Q(i).
a(n) = 16*a(n-4) - a(n-8) for n>7. - Colin Barker, Jul 21 2015
G.f.: -(x+1)*(x^2-x-1)*(x^4+3*x^2+1) / (x^8-16*x^4+1). - Colin Barker, Jul 21 2015

A259597 Numerators of the other-side convergents to sqrt(7).

Original entry on oeis.org

3, 5, 8, 13, 45, 82, 127, 209, 717, 1307, 2024, 3331, 11427, 20830, 32257, 53087, 182115, 331973, 514088, 846061, 2902413, 5290738, 8193151, 13483889, 46256493, 84319835, 130576328, 214896163, 737201475, 1343826622, 2081028097, 3424854719, 11748967107

Offset: 0

Clark Kimberling, Jul 20 2015

Suppose that a positive irrational number r has continued fraction [a(0), a(1), ... ]. Define sequences p(i), q(i), P(i), Q(i) from the numerators and denominators of finite continued fractions as follows:
p(i)/q(i) = [a(0), a(1), ... a(i)] and P(i)/Q(i) = [a(0), a(1), ..., a(i) + 1]. The fractions p(i)/q(i) are the convergents to r, and the fractions P(i)/Q(i) are introduced here as the "other-side convergents" to
r, because p(2k)/q(2k) < r < P(2k)/Q(2k) and P(2k+1)/Q(2k+1) < r < p(2k+1)/q(2k+1), for k >= 0. The closeness of P(i)/Q(i) to r is indicated by |r - P(i)/Q(i)| < |p(i)/q(i) - P(i)/Q(i)| = 1/(q(i)Q(i)), for i >= 0.

			For r = sqrt(7), 3, 5/2, 8/3, 13/5, 45/17, 82/31, 127/48. A comparison of convergents with other-side convergents:
i  p(i)/q(i)            P(i)/Q(i)    p(i)*Q(i)-P(i)*q(i)
0     2/1   < sqrt(7) <    3/1               -1
1     3/1   > sqrt(7) >    5/2                1
2     5/2   < sqrt(7) <    8/3               -1
3     8/3   > sqrt(7) >   13/5                1
4     37/14 < sqrt(7) <   45/17              -1
5     45/17 > sqrt(7) >   83/31               1

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,16,0,0,0,-1).

Cf. A041008, A041009, A259596 (denominators).

r = Sqrt[7]; a[i_] := Take[ContinuedFraction[r, 35], i];
b[i_] := ReplacePart[a[i], i -> Last[a[i]] + 1];
t = Table[FromContinuedFraction[b[i]], {i, 1, 35}]
v = Numerator[t]
LinearRecurrence[{0,0,0,16,0,0,0,-1},{3,5,8,13,45,82,127,209},40] (* Harvey P. Dale, Jan 15 2017 *)

Vec((x^7-x^6+2*x^5-3*x^4+13*x^3+8*x^2+5*x+3)/(x^8-16*x^4+1) + O(x^50)) \\ Colin Barker, Jul 21 2015

p(i)*Q(i) - P(i)*q(i) = (-1)^(i+1), for i >= 0, where a(i) = Q(i).
a(n) = 16*a(n-4) - a(n-8) for n>7. - Colin Barker, Jul 21 2015
G.f.: (x^7-x^6+2*x^5-3*x^4+13*x^3+8*x^2+5*x+3) / (x^8-16*x^4+1). - Colin Barker, Jul 21 2015

cp's OEIS Frontend

A010465 Decimal expansion of square root of 7.

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A041006 Numerators of continued fraction convergents to sqrt(6).

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A041014 Numerators of continued fraction convergents to sqrt(11).

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A042936 Numerators of continued fraction convergents to sqrt(1000).

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A041009 Denominators of continued fraction convergents to sqrt(7).

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A041010 Numerators of continued fraction convergents to sqrt(8).

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A042934 Numerators of continued fraction convergents to sqrt(999).

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A266698 x-values of solutions to the Diophantine equation x^2 - 7*y^2 = 2.

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