A020806 -id:A020806 - cp's OEIS frontend

A144472 Negative values along the main diagonal of the array defined by A020806 and its differences.

-1, 2, 9, 13, 31, 57, 119, 233, 471, 937, 1879, 3753, 7511, 15017, 30039, 60073, 120151, 240297, 480599, 961193, 1922391, 3844777, 7689559, 15379113, 30758231, 61516457, 123032919, 246065833, 492131671, 984263337, 1968526679, 3937053353, 7874106711

Offset: 1

Paul Curtz, Oct 10 2008, Oct 14 2008

			A020806 and its repeated differences in the next rows start as follows:
..1,..4,..2,..8,..5,..7,..1,..4,..2,..8, <- A020806
..3,.-2,..6,.-3,..2,.-6,..3,.-2,..6,.-3, <- A131969
.-5,..8,.-9,..5,.-8,..9,.-5,..8,.-9,..5,
.13,-17,.14,-13,.17,-14,.13,-17,.14,-13,
-30,.31,-27,.30,-31,.27,-30,.31,-27,.30,
.61,-58,.57,-61,.58,-57,.61,-58,.57,-61,
The diagonal is 1,-2,-9,-13,-31,... which yields a(n) after signs are flipped.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,2).

Join[{-1}, LinearRecurrence[{1, 2}, {2, 9}, 40]] (* Jean-François Alcover, Nov 06 2017 *)

Vec(-x*(1 - 3*x - 9*x^2) / ((1 + x)*(1 - 2*x)) + O(x^50)) \\ Colin Barker, Nov 06 2017

a(n+1) - 2*a(n) = (-1)^n*A010716(n), n>1, period 2.
G.f.: x*(1-3*x-9*x^2) / ((1+x)*(2*x-1)). - R. J. Mathar, Oct 24 2008
a(n) = 11*2^(n-2)/3 - 5*(-1)^n/3, n>1. - R. J. Mathar, Oct 24 2008
From Colin Barker, Nov 06 2017: (Start)
a(n) = (11*2^n - 20) / 12 for n>1 and even.
a(n) = (11*2^n + 20) / 12 for n>1 and odd.
a(n) = a(n-1) + 2*a(n-2) for n>3.
(End)

Edited and extended by R. J. Mathar, Oct 24 2008

A257581 Continued square root map applied to the sequence (1,4,2,8,5,7) repeated (A020806).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane, May 02 2015

The continued square root map applied to a sequence b(1), b(2), b(3), ... is defined to be the number sqrt(b(1)+sqrt(b(2)+sqrt(b(3)+sqrt(b(4)+...)))).

			1.8734951093713154879193475...

Hiroaki Yamanouchi, Table of n, a(n) for n = 1..400
Herman P. Robinson, The CSR Function, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC35-3 to PC35-4. Annotated and scanned copy.

Cf. A020806.

a(27)-a(87) from Hiroaki Yamanouchi, May 03 2015

A131969 First differences of A020806.

Original entry on oeis.org

3, -2, 6, -3, 2, -6, 3, -2, 6, -3, 2, -6, 3, -2, 6, -3, 2, -6, 3, -2, 6, -3, 2, -6, 3, -2, 6, -3, 2, -6, 3, -2, 6, -3, 2, -6, 3, -2, 6, -3, 2, -6, 3, -2, 6, -3, 2, -6, 3, -2, 6, -3, 2, -6, 3, -2, 6, -3, 2, -6, 3, -2, 6, -3, 2, -6, 3, -2, 6, -3, 2, -6, 3, -2, 6, -3, 2, -6

Offset: 0

Paul Curtz, Oct 06 2007

sign

a(n)=11*(-1)^n/3-2*A010892(n)/3+7*A010892(n-1)/3, n>0. a(n+6)=a(n). G.f.: (3-2x+6x^2)/((1+x)(1-x+x^2)). [From R. J. Mathar, Oct 24 2008]

A144471 Inverse binomial transform of A020806.

Original entry on oeis.org

1, 3, -5, 13, -30, 61, -119, 234, -467, 937, -1878, 3757, -7511, 15018, -30035, 60073, -120150, 240301, -480599, 961194, -1922387, 3844777, -7689558, 15379117, -30758231, 61516458, -123032915, 246065833, -492131670, 984263341, -1968526679, 3937053354, -7874106707

Offset: 0

Paul Curtz, Oct 10 2008

sign

Index entries for linear recurrences with constant coefficients, signature (-3,-3,-2)

Cf. A141425, A141532.

Digits := 200 ; read("transforms") ; read("transforms3") ; x := 1/7 ; L := CONSTTOLIST(x) ; BINOMIALi(L) ; # R. J. Mathar, Sep 07 2009

LinearRecurrence[{-3,-3,-2},{1,3,-5,13},40] (* Harvey P. Dale, Nov 11 2017 *)

|a(n+1)| - 2*|a(n)| = -A117378(n-1) = A117378(n+2), n>0.
a(n) = -3*a(n-1) - 3*a(n-2) - 2*a(n-3), n > 3.
G.f.: (6*x+7*x^2+9*x^3+1) / ((2*x+1) * (1+x+x^2)). - R. J. Mathar, Sep 07 2009

Edited and extended by R. J. Mathar, Sep 07 2009

A153130 Period 6: repeat [1, 2, 4, 8, 7, 5].

Original entry on oeis.org

1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 7, 5

Offset: 0

Paul Curtz, Dec 19 2008

Digital root of 2^n.
A regular version of Pitoun's sequence: a(n) = A029898(n+1).
Also obtained from permutations of A141425, A020806, A070366, A153110, A153990, A154127, A154687, or A154815.
This sequence and its (again period 6) repeated differences produce the table:
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4, -1, -2, -4,  1,  2,  4, -1, -2, ...
    1,  2, -5, -1, -2,  5,  1,  2, -5, -1, -2, ...
    1, -7,  4, -1,  7, -4,  1, -7,  4, -1,  7, ...
   -8, 11, -5,  8,-11,  5, -8, 11, -5,  8,-11, ...
   19,-16, 13,-19, 16,-13, 19,-16, 13,-19, 16, ...
  -35, 29,-32, 35,-29, 32,-35, 29,-32, 35,-29, ...
   64,-61, 67,-64, 61,-67, 64,-61, 67,-64, 61, ...
If each entry of this table is read modulo 9 we obtain the very regular table:
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
    1,  2,  4,  8,  7,  5,  1,  2,  4,  8,  7, ...
Also the decimal expansion of the constant 125/1001. - R. J. Mathar, Jan 23 2009
Digital root of the powers of any number congruent to 2 mod 9. - Alonso del Arte, Jan 26 2014

Cecil Balmond, Number 9: The Search for the Sigma Code. Munich, New York: Prestel (1998): 203.

Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A004000, A007953, A010734, A010888, A029898, A030132, A082365, A145389, A189510.
Cf. digital roots of powers of c mod 9: c = 4, A100402; c = 5, A070366; c = 7, A070403; c = 8, A010689.
Cf. A020806, A070366, A141425, A153110, A153990, A154127, A154687, A154815.

&cat [[1, 2, 4, 8, 7, 5]^^30]; // Wesley Ivan Hurt, Jul 05 2016

seq(op([1, 2, 4, 8, 7, 5]), n=0..40); # Wesley Ivan Hurt, Jul 05 2016

Flatten[Table[{1, 2, 4, 8, 7, 5}, {20}]] (* Paul Curtz, Dec 19 2008 *)
Table[Mod[2^n, 9], {n, 0, 99}] (* Alonso del Arte, Jan 26 2014 *)

a(n)=lift(Mod(2,9)^n) \\ Charles R Greathouse IV, Apr 21 2015

a(n) + a(n+3) = 9 = A010734(n).
G.f.: (1+x+2x^2+5x^3)/((1-x)(1+x)(1-x+x^2)). - R. J. Mathar, Jan 23 2009
a(n) = A082365(n) mod 9. - Paul Curtz, Mar 31 2009
a(n) = -1/2*cos(Pi*n) - 3*cos(1/3*Pi*n) - 3^(1/2)*sin(1/3*Pi*n) + 9/2. - Leonid Bedratyuk, May 13 2012
a(n) = A010888(A004000(n+1)). - Ivan N. Ianakiev, Nov 27 2014
From Wesley Ivan Hurt, Apr 20 2015: (Start)
a(n) = a(n-6) for n>5.
a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
a(n) = (2+3*(n-1 mod 3))*(n mod 2) + (1+3*(-n mod 3))*(n-1 mod 2). (End)
a(n) = 2^n mod 9. - Nikita Sadkov, Oct 06 2018
From Stefano Spezia, Mar 20 2025: (Start)
E.g.f.: 4*cosh(x) - exp(x/2)*(3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2)) + 5*sinh(x).
a(n) = A007953(2*a(n-1)) = A010888(2*a(n-1)). (End)

Edited by R. J. Mathar, Apr 09 2009

A057357 a(n) = floor(3*n/7).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 9, 9, 10, 10, 11, 11, 12, 12, 12, 13, 13, 14, 14, 15, 15, 15, 16, 16, 17, 17, 18, 18, 18, 19, 19, 20, 20, 21, 21, 21, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 27, 27, 27, 28, 28, 29, 29, 30, 30, 30, 31, 31, 32, 32

Offset: 0

Mitch Harris

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

This sequence relates to 3/7 = 0.42857142... (essentially given by A020806). It differs from the Beatty sequence A308358 for sqrt(3)/4 = 0.43301270... = A120011.

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

G. C. Greubel, Table of n, a(n) for n = 0..5000
N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).
Index to sequences related to Beatty sequences.

Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

Cf. A020806, A193535, A308358.

[Floor(3*n/7): n in [0..50]]; // G. C. Greubel, Nov 02 2017

Floor[(3*Range[0,80])/7] (* Harvey P. Dale, Aug 22 2013 *)

a(n)=3*n\7 \\ Charles R Greathouse IV, Sep 02 2015

G.f.: (1+x^2+x^4)*x^3/((1-x)*(1-x^7)) - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002
for all m>=0 a(7m)=0 mod 3; a(7m+1)=0 mod 3; a(7m+2)= 0 mod 3; a(7m+3) = 1 mod 3; a(5m+4) = 1 mod 3; a(7m+5) = 2 mod 3; a(7m+6) = 2 mod 3 - Bruce Corrigan (scentman(AT)myfamily.com), Jul 03 2002
Sum_{n>=3} (-1)^(n+1)/a(n) = log(2)/3 (A193535). - Amiram Eldar, Sep 30 2022

A068028 Decimal expansion of 22/7.

Original entry on oeis.org

3, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4, 2, 8, 5, 7, 1, 4

Offset: 1

Nenad Radakovic, Mar 22 2002

This is an approximation to Pi. It is accurate to 0.04025%.
Consider the recurring part of 22/7 and the sequences R(i) = 2, 1, 4, 2, 3, 0, 2, ... and Q(i) = 1, 4, 2, 8, 5, 7, 1, .... For i > 0, let X(i) = 10*R(i) + Q(i). Then Q(i+1) = floor(X(i)/Y); R(i+1) = X(i) - Y*Q(i+1); here Y=5; X(0)=X=7. Note 1/7 = 7/49 = X/(10*Y-1). Similar comment holds elsewhere. If we consider the sequences R(i) = 3, 2, 3, 5, 5, 1, 4, 0, 6, 4, 6, 3, 4, 3, 1, 1, 5, 2, 6, 0, 2, 0, 3, ... and Q(i) = A021027, we have X=3; Y=7 (attributed to Vedic literature). - K.V.Iyer, Jun 16 2010, Jun 18 2010
The sequence of convergents of the continued fraction of Pi begins [3, 22/7, 333/106, 355/113, 103993/33102, ...]. 22/7 is the second convergent. The summation 240*Sum_{n >= 1} 1/((4*n+1)*(4*n+2)*(4*n+3)*(4*n+5)(4*n+6)*(4*n+7)) = 22/7 - Pi shows that 22/7 is an over-approximation to Pi. - Peter Bala, Oct 12 2021

John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 187, 239.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §3.6 The Quest for Pi and §13.3 Solving Triangles, pp. 90, 479.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 49.

D. Castellanos, The ubiquitous pi, Math. Mag., 61 (1988), 67-98 and 148-163. - _N. J. A. Sloane_, Mar 24 2012
D. P. Dalzell, On 22/7, J. London Math. Soc. 19, 133-134, 1944.
Dale Winham, Facts about Pi.
Index entries for sequences related to the number Pi.
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A068079, A068089, A002485, A002486, A046965, A046947.

I:=[3,1,4,2,8]; [n le 5 select I[n] else Self(n-1)-Self(n-3)+Self(n-4): n in [1..100]]; // Vincenzo Librandi, Mar 27 2015

CoefficientList[Series[(3 - 2 x + 3 x^2 + x^3 + 4 x^4) / ((1 - x) (1 + x) (1 - x + x^2)), {x, 0, 100}], x] (* Vincenzo Librandi, Mar 27 2015 *)
Join[{3},LinearRecurrence[{1, 0, -1, 1},{1, 4, 2, 8},104]] (* Ray Chandler, Aug 26 2015 *)
RealDigits[22/7,10,120][[1]] (* Harvey P. Dale, Oct 04 2021 *)

a(0)=3, a(n) = floor(714285/10^(5-(n mod 6))) mod 10. - Sascha Kurz, Mar 23 2002 [corrected by Jason Yuen, Aug 18 2024]
Equals 100*A021018 - 4 = 3 + A020806. - R. J. Mathar, Sep 30 2008
For n>1 a(n) = A020806(n-2) (note offset=0 in A020806 and offset=1 in A068028). - Zak Seidov, Mar 26 2015
G.f.: x*(3-2*x+3*x^2+x^3+4*x^4)/((1-x)*(1+x)*(1-x+x^2)). - Vincenzo Librandi, Mar 27 2015

More terms from Sascha Kurz, Mar 23 2002

Alternative to broken link added by R. J. Mathar, Jun 18 2010

A351474 Numbers m such that the largest digit in the decimal expansion of 1/m is 8.

Original entry on oeis.org

7, 12, 14, 26, 28, 35, 48, 54, 55, 56, 63, 65, 70, 72, 78, 79, 93, 117, 120, 123, 125, 128, 140, 175, 176, 186, 192, 195, 205, 224, 239, 259, 260, 264, 280, 296, 312, 318, 328, 350, 372, 416, 432, 438, 448, 465, 480, 540, 542, 546, 548, 550, 555, 560, 572, 584, 594, 630, 632, 650, 675

Offset: 1

Bernard Schott and Robert G. Wilson v, Feb 19 2022

If k is a term, 10*k is also a term. First few primitive terms are 7, 12, 14, 26, 28, 35, 48, 54, 55, 56, 63, 65, 72, ...

The seven primes up to 2.7*10^8 are 7, 79, 239, 62003, 538987, 35121409, 265371653 (see comments in A333237, example section and Crossrefs).

			As 1/7 = 0.142857142857142857..., 7 is a term.
As 1/26 = 0.0384615384615384615..., 26 is another term.

Index entries for sequences related to decimal expansion of 1/n.

Similar with largest digit k: A333402 (k=1), A341383 (k=2), A350814 (k=3), A351470 (k=4), A351471 (k=5), A351472 (k=6), A351473 (k=7), this sequence (k=8), A333237 (k=9).
Cf. A333236.
Decimal expansion of: A020806 (1/7), A021058 (1/54), A021060 (1/56), A021067 (1/63), A021069 (1/65), A021083 (1/79), A021097 (1/93).

f[n_] := Union[ Flatten[ RealDigits[ 1/n][[1]] ]]; Select[Range@1500000, Max@ f@# == 8 &]

isok(m) = my(m2=valuation(m, 2), m5=valuation(m, 5)); vecmax(digits(floor(10^(max(m2,m5) + znorder(Mod(10, m/2^m2/5^m5))+1)/m))) == 8; \\ Michel Marcus, Feb 26 2022

from itertools import count, islice
from sympy import multiplicity, n_order
def A351474_gen(startvalue=1): # generator of terms >= startvalue
    for a in count(max(startvalue,1)):
        m2, m5 = (~a&a-1).bit_length(), multiplicity(5,a)
        k, m = 10**max(m2,m5), 10**n_order(10,a//(1<A351474_list = list(islice(A351474_gen(),20)) # Chai Wah Wu, May 02 2023

A333236(a(n)) = 8.

A021069 Decimal expansion of 1/65.

Original entry on oeis.org

0, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5, 3, 8, 4, 6, 1, 5

Offset: 0

N. J. A. Sloane

Without the leading 0 also the decimal expansion of 2/13.

			0.0153846153846...  - _Natan Arie Consigli_, Sep 18 2016

G. C. Greubel, Table of n, a(n) for n = 0..10000
Christian Táfula, Knights are 24/13 times faster than the king, arXiv:2405.19589 [math.CO], 2024.
Index entries for linear recurrences with constant coefficients, signature (1,0,-1,1).

Cf. A010734, A020806, A021017, A021095, A128834, A130806, A144472.

Magma
```
1./65; // G. C. Greubel, Jan 15 2018
```

Digits:=100: evalf(1/65); # Wesley Ivan Hurt, Sep 20 2016

Join[ConstantArray[0, Abs@ Last@ #], First@ #] &@ RealDigits[N[1/65, 120]] (* Michael De Vlieger, Sep 06 2016 *)

1./65. /* Michel Marcus, Aug 22 2016 */

Equals 2 - 24/13. See Táfula link. - Michel Marcus, May 31 2024

G.f.: x*(1 + 4*x - 2*x^2 + 6*x^3)/((1 - x)*(1 + x)*(1 - x + x^2)). - Stefano Spezia, Apr 30 2025

A242824 Primes formed by the initial digits of the decimal expansion of 1/7, starting at the first nonzero digit in the expansion.

Original entry on oeis.org

1428571, 1428571428571428571428571

Offset: 1

Felix Fröhlich, May 23 2014

Next term has 355 digits.

All terms are of the form 6x+1; a(4) has 823 digits; and there are no further terms up to and including 10000 digits. - Harvey P. Dale, Oct 03 2018

Amiram Eldar, Table of n, a(n) for n = 1..4

Cf. A020806, A241217.

Corresponding sequences for 1/k: A093676 (k=12), A242826 (k=13), A242827 (k=14), A242828 (k=17), A242833 (k=19).

Select[Table[FromDigits[PadRight[{},6n+1,{1,4,2,8,5,7}]],{n,200}],PrimeQ](* Harvey P. Dale, Oct 03 2018 *)

cp's OEIS Frontend

A144472 Negative values along the main diagonal of the array defined by A020806 and its differences.

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A257581 Continued square root map applied to the sequence (1,4,2,8,5,7) repeated (A020806).

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A131969 First differences of A020806.

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A144471 Inverse binomial transform of A020806.

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A153130 Period 6: repeat [1, 2, 4, 8, 7, 5].

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A057357 a(n) = floor(3*n/7).

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A068028 Decimal expansion of 22/7.

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A351474 Numbers m such that the largest digit in the decimal expansion of 1/m is 8.

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