A130784 -id:A130784 - cp's OEIS frontend

A158289 Period 18 zigzag sequence: repeat [0,1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1].

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

Offset: 0

Jaroslav Krizek, Mar 15 2009

A toothed or zigzag sequence.
Sequence contains only numbers 0..9; abs(a(n+1)-a(n)) = 1.
Decimal expansion of 12345679/1000000001. - Elmo R. Oliveira, Feb 20 2024

Arkadiusz Wesolowski, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,-1,1).

Cf. A068073 (repeat 1,2,3,2), A028356 (repeat 1,2,3,4,3,2), A130784 (repeat 1,3,2).

Period k zigzag sequences: A000035 (k=2), A007877 (k=4), A260686 (k=6), A266313 (k=8), A271751 (k=10), A271832 (k=12), A279313 (k=14), A279319 (k=16), this sequence (k=18).

[ s lt 9 select r else 9-r where r is n mod 9 where s is n mod 18: n in [0..104] ]; // Klaus Brockhaus, Sep 07 2009

S:=[]; a:=0; for n in [0..104] do Append(~S, a); if n mod 18 eq 0 then d:=1; else if n mod 9 eq 0 then d:=-1; end if; end if; a+:=d; end for; S; // Klaus Brockhaus, Sep 07 2009

&cat[[0,1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1]: n in [0..5]]; // Vincenzo Librandi, Jul 26 2015

a[n_] := If[m = Mod[n, 18]; m <= 9, m, 18-m]; Table[a[n], {n, 0, 85}] (* Jean-François Alcover, Jul 19 2013 *)
PadRight[{}, 100, {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 7, 6, 5, 4, 3, 2, 1}] (* Vincenzo Librandi, Jul 26 2015 *)

a(n)=abs(n-round(n/18)*18) \\ M. F. Hasler, Jul 27 2015

a(18*k+j) = a(18*(k+1)-j) = j for k >= 0, j = 0..9.
G.f.: x*(1+x+x^2)*(1+x^3+x^6)/((1-x)*(1+x)*(1-x+x^2)*(1-x^3+x^6)). - Klaus Brockhaus, Sep 07 2009
a(n) = Sum_{i=0..n-1} (-1)^floor(i/9). - Wesley Ivan Hurt, Jul 25 2015
a(n) = abs(n - 18*round(n/18)). - Wesley Ivan Hurt, Dec 10 2016
a(n) = a(n-18) for n >= 18. - Wesley Ivan Hurt, Sep 07 2022

Edited and extended by Klaus Brockhaus, Sep 07 2009

A274921 Spiral constructed on the nodes of the triangular net in which each new term is the least positive integer distinct from its neighbors.

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 3, 1, 3, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 3, 2, 1, 3, 2, 3, 1, 2, 3, 2, 1, 3, 2, 3, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 1, 2, 1, 3, 2, 1, 3, 1, 2, 3, 1, 2, 1, 3, 2, 1, 3, 1, 2, 3, 1, 2, 3, 2, 1, 3, 2, 1, 2, 3, 1, 2, 3, 1, 3, 2, 1

Offset: 0

Omar E. Pol, Jul 11 2016

nonn

The structure of the spiral has the following properties:
1) Every 1 is surrounded by three equidistant 2's and three equidistant 3's.
2) Every 2 is surrounded by three equidistant 1's and three equidistant 3's.
3) Every 3 is surrounded by three equidistant 1's and three equidistant 2's.
4) Diagonals are periodic sequences with period 3 (A010882 and A130784).
From Juan Pablo Herrera P., Nov 16 2016: (Start)
5) Every hexagon with a 1 in its center is the same hexagon as the one in the middle of the spiral.
6) Every triangle whose number of numbers is divisible by 3 has the same number of 1's, 2's, and 3's. For example, a triangle with 6 numbers, has two 1's, two 2's, and two 3's. (End)
a(n) = a(n-2) if n > 2 is in A014591, otherwise a(n) = 6 - a(n-1)-a(n-2). - Robert Israel, Sep 15 2017

			Illustration of initial terms as a spiral:
.
.                3 - 1 - 2 - 3 - 1 - 2
.               /                     \
.              1   2 - 3 - 1 - 2 - 3   1
.             /   /                 \   \
.            2   3   1 - 2 - 3 - 1   2   3
.           /   /   /             \   \   \
.          3   1   2   3 - 1 - 2   3   1   2
.         /   /   /   /         \   \   \   \
.        1   2   3   1   2 - 3   1   2   3   1
.       /   /   /   /   /     \   \   \   \   \
.      2   3   1   2   3   1 - 2   3   1   2   3
.       \   \   \   \   \         /   /   /   /
.        1   2   3   1   2 - 3 - 1   2   3   1
.         \   \   \   \             /   /   /
.          3   1   2   3 - 1 - 2 - 3   1   2
.           \   \   \                 /   /
.            2   3   1 - 2 - 3 - 1 - 2   3
.             \   \                     /
.              1   2 - 3 - 1 - 2 - 3 - 1
.               \
.                3 - 1 - 2 - 3 - 1 - 2
.

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A001399, A010882, A130784, A253186, A274820, A274821, A274920, A275606, A275610.

A[0]:= 1: A[1]:= 2: A[2]:= 3:
b:= 3: c:= 2: d:= 2: e:= 1: f:= 1:
for n from 3 to 200 do
  if n = b then
     r:= b; b:= c + d - f + 1; f:= e; e:= d; d:= c; c:= r;
     A[n]:= A[n-2];
  else
     A[n]:= 6 - A[n-1] - A[n-2];
  fi
od:
seq(A[i],i=0..200); # Robert Israel, Sep 15 2017

a(n) = A274920(n) + 1.

A240438 Greatest minimal difference between numbers of adjacent cells in a regular hexagonal honeycomb of order n with cells numbered from 1 through the total number of cells, the order n corresponding to the number of cells on one side of the honeycomb.

Original entry on oeis.org

0, 1, 5, 11, 18, 28, 40, 53, 69, 87, 106, 128, 152, 177, 205, 235, 266, 300, 336, 373, 413, 455, 498, 544, 592, 641, 693, 747, 802, 860, 920, 981, 1045, 1111, 1178, 1248, 1320, 1393, 1469, 1547, 1626, 1708, 1792, 1877, 1965, 2055, 2146, 2240, 2336, 2433, 2533, 2635

Offset: 1

Jörg Zurkirchen, Apr 05 2014

Difference table of a(n), with a(0)=0 and offset=0:
0,   0,  1,  5, 11, 18, 28, 40, 53, 69, ...
0,   1,  4,  6,  7, 10, 12, 13, 16, 18, ...   =  A047234(n+1)
1,   3,  2,  1,  3,  2,  1,  3,  2,  1, ...   =  A130784
2,  -1, -1,  2, -1, -1,  2, -1, -1,  2, ...   = -A131713(n+1)
-3,  0,  3, -3,  0,  3, -3,  0,  3, -3; ...   =  A099838(n+4)
3,   3, -6,  3,  3, -6,  3,  3, -6,  3, ...
0,  -9,  9,  0, -9,  9,  0, -9,  9,  0, ...
-9, 18, -9, -9, 18, -9, -9, 18, -9, -9, ...
First column: see A057682. - Paul Curtz, Nov 11 2014
Diameter of the chamber graph Γ(Alt(2n+1)). Definition of this graph:
Each vertex v is a sequence (v[1],v[2],...,v[n]) of length n, where each v[i] is a 2-subset of {1,2,...,2n+1} and v[i] and v[j] are disjoint unless i=j.
Vertices u and v are connected iff either:
u and v are identical except for their first elements u[1] and v[1], or
u and v are identical except for some i for which u[i]=v[i+1] and v[i]=u[i+1] - Tim Crinion, 17 Feb 2019

			For n = 3 an example of a honeycomb with the greatest minimal difference of a(3) = 5 is:
.         __
.      __/ 7\__
.   __/15\__/13\__
.  / 4\__/ 2\__/ 1\
.  \__/10\__/ 8\__/
.  /18\__/16\__/14\
.  \__/ 5\__/ 3\__/
.  /12\__/11\__/ 9\
.  \__/19\__/17\__/
.     \__/ 6\__/
.        \__/
.

22ème Championnat des jeux mathématiques et logiques - 1/4 de finale individuels 2008, problème 18, «Les ruches d’Abella»

Vincenzo Librandi, Table of n, a(n) for n = 1..1000 (first 100 terms from Jörg Zurkirchen)
Tim Crinion, Chamber Graphs of some geometries related to the Petersen Graph, 2013.
Fédération Suisse des Jeux Mathématiques, 22nd Championship of Mathematical and Logical Games - Quarter Final 2008, 18 problems in French; problem number 18 was decisive to creating this sequence. See following pdf for an English version of problem 18.
Jörg Zurkirchen, Honeycomb.pdf
Index entries for linear recurrences with constant coefficients, signature (2, -1, 1, -2, 1).

Cf. A042965, A047234, A057682, A099838, A130784, A131713.

[n*(n-1)-Floor((n+1)/3): n in [1..60]]; // Vincenzo Librandi, Nov 12 2014

A240438:=n->n*(n-1)-floor((n+1)/3); seq(A240438(n), n=1..50); # Wesley Ivan Hurt, Apr 08 2014

Table[n (n - 1) - Floor[(n + 1)/3], {n, 50}] (* Wesley Ivan Hurt, Apr 08 2014 *)
CoefficientList[Series[x (x + 1) (2 x + 1) / ((1 - x)^3 (x^2 + x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Nov 12 2014 *)
LinearRecurrence[{2, -1, 1, -2, 1},{0, 1, 5, 11, 18},52] (* Ray Chandler, Sep 24 2015 *)

a(n) = n*(n-1)-floor((n+1)/3).
G.f.: -x^2*(x+1)*(2*x+1) / ((x-1)^3*(x^2+x+1)). - Colin Barker, Apr 08 2014
a(n+3) = a(n) + 6*n+5. - Paul Curtz, Nov 11 2014
a(n) = n^2 - (A042965(n+1)=0, 1, 3, 4, ...). - Paul Curtz, Nov 11 2014
a(n+1) = a(n) + A047234(n+1). - Paul Curtz, Nov 11 2014

A130785 Sequence identical to its third differences: a(n+3) = 3a(n+2)-3a(n+1)+2a(n), with a(0)=1, a(1)=4, a(2)=9.

Original entry on oeis.org

1, 4, 9, 17, 32, 63, 127, 256, 513, 1025, 2048, 4095, 8191, 16384, 32769, 65537, 131072, 262143, 524287, 1048576, 2097153, 4194305, 8388608, 16777215, 33554431, 67108864, 134217729, 268435457, 536870912, 1073741823, 2147483647, 4294967296, 8589934593

Offset: 0

Paul Curtz, Jul 15 2007

From R. J. Mathar, Nov 22 2007: (Start)
Sequences which equal the sequence of their d-th differences obey linear recurrences with constant binomial coefficients of the form Sum_{i=0..d} binomial(d,d-i)*(-1)^i*a(n-i) = a(n-d).
If d is even, this simplifies to Sum_{i=0..d-1} binomial(d,d-i)*(-1)^i*a(n-i) = 0.
This binding of d (d odd) or d-1 (d even) consecutive terms by the recurrences leaves d or d-1, respectively, free parameters to choose a(0),a(1),...,a(d) or a(0),a(1),...,a(d-1), respectively, which ultimately define the individual sequence.
The generating functions are
d=2: a(0)/(1-2*x).
d=3: (1/3)*(-a(0) + a(1) - a(2))/(-1+2*x) + (1/3)*(-4*a(0)*x - x*a(2) + 4*a(1)*x - a(2) + 2*a(0) + a(1))/(x^2-x+1).
d=4: (1/2)*(-2*a(0) + 2*a(1) - a(2))/(-1+2*x) + (1/2)*(2*a(1)*x - 4*a(0)*x - a(2) + 2*a(1))/(1 - 2*x + 2*x^2).
In the present sequence we have d=3 and g.f. = (x-1)/(x^2-x+1) - 2/(-1+2*x). (End)
Also binomial transform of A130784. a(n) = 2^(n+1) + A010892(n+4).
Recurrence in shorter form: a(n) = 2*a(n) + periodically extended [2, 1, -1, -2, -1, 1].
See A130750, A130752, A130755 for other examples of d=3 sequences, A130781 for an example of d=4.

			Triangle of sequence and 1st, 2nd, 3rd differences:
  1   4   9  17  32  63 127 256 513
    3   5   8  15  31  64 129 257
      2   3   7  16  33  65 128
        1   4   9  17  32  63 ... equal to first row

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

d = 3; nmax = 20; a[n_ /; n < d] := (n+1)^2; seq = Table[a[n], {n, 0, nmax}]; seq /. Solve[ Thread[ Take[seq, nmax - d + 1] == Differences[seq, d]]] // First (* Jean-François Alcover, Nov 07 2013 *)
LinearRecurrence[{3, -3, 2},{1, 4, 9},21] (* Ray Chandler, Sep 23 2015 *)
Table[2^(n + 1) - Cos[(2 n + 1) Pi/6] 2/Sqrt[3], {n, 0, 32}] (* Vladimir Reshetnikov, Oct 15 2017 *)

a(n) = 2^(n+1) - cos((2*n+1)*Pi/6) * 2/sqrt(3). - Vladimir Reshetnikov, Oct 15 2017

G.f.: (1+x)/((1-2*x)*(1-x+x^2)). - Joerg Arndt, Oct 16 2017

Edited and extended by R. J. Mathar, Nov 22 2007

A131756 Period 3: repeat [2, -1, 3].

Original entry on oeis.org

2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3, 2, -1, 3

Offset: 0

Paul Curtz, Oct 04 2007

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

Cf. A130784.

&cat [[2, -1, 3]^^30]; // Wesley Ivan Hurt, Jul 01 2016

seq(op([2, -1, 3]), n=0..50); # Wesley Ivan Hurt, Jul 01 2016

PadRight[{}, 100, {2, -1, 3}] (* Wesley Ivan Hurt, Jul 01 2016 *)

a(n)=[2,-1,3][1+n%3] \\ Jaume Oliver Lafont, Mar 24 2009

a(n) = 4/3+2/3*cos(2/3*Pi*n)-4/3*3^(1/2)*sin(2/3*Pi*n). - R. J. Mathar, Nov 15 2007
G.f.: (2-x+3*x^2)/(1-x^3). - Jaume Oliver Lafont, Mar 24 2009
a(n) = a(n-3) for n>2. - Wesley Ivan Hurt, Jul 01 2016

A134977 Period 6: repeat [1, 4, 2, 3, 0, 2].

Original entry on oeis.org

1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2, 1, 4, 2, 3, 0, 2

Offset: 0

Paul Curtz, Feb 04 2008

Northwest diagonal sums of A134658, omitting row 0.

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

Cf. A010892, A119910, A130784, A131756, A134658.

&cat[[1, 4, 2, 3, 0, 2]^^20]; // Wesley Ivan Hurt, Jun 18 2016

A134977:=n->[1, 4, 2, 3, 0, 2][(n mod 6)+1]: seq(A134977(n), n=0..100); # Wesley Ivan Hurt, Jun 18 2016

Flatten[Table[{1, 4, 2, 3, 0, 2}, {20}]] (* Wesley Ivan Hurt, Jun 18 2016 *)
PadRight[{}, 100, {1, 4, 2, 3, 0, 2}] (* Vincenzo Librandi, Jun 19 2016 *)

O.g.f.: -1/(x+1)-2/(x-1)+x/(x^2-x+1). a(n) = 2-(-1)^n+A010892(n-1). - R. J. Mathar, Feb 08 2008
From Wesley Ivan Hurt, Jun 18 2016: (Start)
a(n) = a(n-1) - a(n-3) + a(n-4) for n>3.
a(n) = (6-3*cos(n*Pi)+2*sqrt(3)*sin(n*Pi/3))/3. (End)

A176977 Decimal expansion of (3+sqrt(37))/7.

Original entry on oeis.org

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

Offset: 1

Klaus Brockhaus, Apr 30 2010

Continued fraction expansion of (3+sqrt(37))/7 is A130784.

			1.29753750432831709842...

Daniel Starodubtsev, Table of n, a(n) for n = 1..10000

Cf. A010491 (decimal expansion of sqrt(37)), A130784 (repeat 1, 3, 2).

RealDigits[(3+Sqrt[37])/7,10,103][[1]] (* Stefano Spezia, May 26 2025 *)

A164360 Period 3: repeat [5, 4, 3].

Original entry on oeis.org

5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3, 5, 4, 3

Offset: 0

Stephen Crowley, Aug 14 2009

From Klaus Brockhaus, May 29 2010: (Start)
Continued fraction expansion of (32+sqrt(1297))/13.
Decimal expansion of 181/333. (End)

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

Cf. A007877 (repeat 0,1,2,1), A068073 (repeat 1,2,3,2), A028356 (repeat 1,2,3,4,3,2), A130784 (repeat 1,3,2), A158289 (repeat 0,1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1).

Cf. A178566 (decimal expansion of (32+sqrt(1297))/13). [Klaus Brockhaus, May 29 2010]

[ n le 3 select 6-n else Self(n-3):n in [1..105] ]; // Klaus Brockhaus, Sep 17 2009

&cat [[5, 4, 3]^^30]; // Wesley Ivan Hurt, Jul 01 2016

seq(op([5, 4, 3]), n=0..50); # Wesley Ivan Hurt, Jul 01 2016

PadRight[{}, 100, {5, 4, 3}] (* Wesley Ivan Hurt, Jul 01 2016 *)

a(n) = 4+(-1)^n*((1/2+I*sqrt(3)/6)*((1+I*sqrt(3))/2)^n+(1/2-I*sqrt(3)/6)*((1-I*sqrt(3))/2)^n). [Corrected by Klaus Brockhaus, Sep 17 2009]
a(n) = 4+(1/3)*sqrt(3)*sin(2*n*Pi/3)+cos(2*n*Pi/3). [Corrected by Klaus Brockhaus, Sep 17 2009]
a(n) = a(n-3) for n > 2, with a(0) = 5, a(1) = 4, a(2) = 3.
G.f.: (5+4*x+3*x^2)/((1-x)*(1+x+x^2)). [Klaus Brockhaus, Sep 17 2009]
E.g.f.: 4*exp(x)+(1/3)*sqrt(3)*exp(-(1/2)*x)*sin((1/2)*x*sqrt(3))+exp(-(1/2)*x)*cos((1/2)*x*sqrt(3)).
a(n) = 4 + A057078(n). - Wesley Ivan Hurt, Jul 01 2016

Edited by Klaus Brockhaus, Sep 17 2009

Offset changed to 0 and formulas adjusted by Klaus Brockhaus, May 18 2010

cp's OEIS Frontend

A158289 Period 18 zigzag sequence: repeat [0,1,2,3,4,5,6,7,8,9,8,7,6,5,4,3,2,1].

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A274921 Spiral constructed on the nodes of the triangular net in which each new term is the least positive integer distinct from its neighbors.

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A240438 Greatest minimal difference between numbers of adjacent cells in a regular hexagonal honeycomb of order n with cells numbered from 1 through the total number of cells, the order n corresponding to the number of cells on one side of the honeycomb.

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A130785 Sequence identical to its third differences: a(n+3) = 3a(n+2)-3a(n+1)+2a(n), with a(0)=1, a(1)=4, a(2)=9.

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A131756 Period 3: repeat [2, -1, 3].

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A134977 Period 6: repeat [1, 4, 2, 3, 0, 2].

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A176977 Decimal expansion of (3+sqrt(37))/7.

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