A169985 -id:A169985 - cp's OEIS frontend

A306417 Number of self-conjugate set partitions of {1, ..., n}.

1, 1, 0, 1, 1, 2, 7, 7, 46, 39, 321

Offset: 0

Gus Wiseman, Feb 14 2019

This sequence counts set partitions fixed under Callan's conjugation operation.

			The  a(3) = 1 through a(7) = 7 self-conjugate set partitions:
  {{12}{3}}  {{13}{24}}  {{123}{4}{5}}  {{135}{246}}    {{13}{246}{57}}
                         {{13}{2}{45}}  {{124}{35}{6}}  {{15}{246}{37}}
                                        {{13}{25}{46}}  {{1234}{5}{6}{7}}
                                        {{14}{2}{356}}  {{124}{3}{56}{7}}
                                        {{14}{236}{5}}  {{134}{2}{5}{67}}
                                        {{14}{25}{36}}  {{14}{2}{3}{567}}
                                        {{145}{26}{3}}  {{14}{23}{57}{6}}

David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.

Cf. A000110, A000126, A000296, A001610, A032032, A052841, A080107, A169985, A306416, A324011, A324012.

A324012 Number of self-complementary set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

Original entry on oeis.org

1, 0, 0, 0, 1, 0, 3, 2, 14, 11, 80, 85, 510

Offset: 0

Gus Wiseman, Feb 12 2019

The complement of a set partition pi of {1, ..., n} is defined as n + 1 - pi (elementwise) on page 3 of Callan. For example, the complement of {{1,5},{2},{3,6},{4}} is {{1,4},{2,6},{3},{5}}. This sequence counts certain self-conjugate set partitions, i.e., fixed points under Callan's conjugation operation.

			The  a(6) = 3 through a(9) = 11 self-complementary set partitions with no singletons or cyclical adjacencies:
  {{135}{246}}    {{13}{246}{57}}  {{1357}{2468}}      {{136}{258}{479}}
  {{13}{25}{46}}  {{15}{246}{37}}  {{135}{27}{468}}    {{147}{258}{369}}
  {{14}{25}{36}}                   {{146}{27}{358}}    {{148}{269}{357}}
                                   {{147}{258}{36}}    {{168}{249}{357}}
                                   {{157}{248}{36}}    {{13}{258}{46}{79}}
                                   {{13}{24}{57}{68}}  {{14}{258}{37}{69}}
                                   {{13}{25}{47}{68}}  {{14}{28}{357}{69}}
                                   {{14}{26}{37}{58}}  {{16}{258}{37}{49}}
                                   {{14}{27}{36}{58}}  {{16}{28}{357}{49}}
                                   {{15}{26}{37}{48}}  {{17}{258}{39}{46}}
                                   {{15}{27}{36}{48}}  {{18}{29}{357}{46}}
                                   {{16}{24}{38}{57}}
                                   {{16}{25}{38}{47}}
                                   {{17}{28}{35}{46}}

David Callan, On conjugates for set partitions and integer compositions, arXiv:math/0508052 [math.CO], 2005.

Cf. A000110, A000126, A000296, A001610, A080107, A169985, A261139, A306417 (all self-conjugate set partitions), A324011 (self-complementarity not required), A324013 (adjacencies allowed), A324014 (singletons allowed), A324015.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
cmp[stn_]:=Union[Sort[Max@@Join@@stn+1-#]&/@stn];
Table[Select[sps[Range[n]],And[cmp[#]==Sort[#],Count[#,{_}]==0,Total[If[First[#]==1&&Last[#]==n,1,0]+Count[Subtract@@@Partition[#,2,1],-1]&/@#]==0]&]//Length,{n,0,10}]

A222334 T(n,k)=Number of binary arrays indicating the locations of trailing edge maxima of a random length-n 0..k array extended with zeros and convolved with 1,1.

Original entry on oeis.org

2, 2, 3, 2, 3, 4, 2, 3, 4, 6, 2, 3, 4, 7, 9, 2, 3, 4, 7, 11, 13, 2, 3, 4, 7, 11, 17, 19, 2, 3, 4, 7, 11, 18, 27, 28, 2, 3, 4, 7, 11, 18, 29, 42, 41, 2, 3, 4, 7, 11, 18, 29, 46, 66, 60, 2, 3, 4, 7, 11, 18, 29, 47, 74, 104, 88, 2, 3, 4, 7, 11, 18, 29, 47, 76, 118, 163, 129, 2, 3, 4, 7, 11, 18, 29, 47

Offset: 1

R. H. Hardin Feb 15 2013

Table starts
....2....2.....2.....2.....2.....2.....2.....2.....2.....2.....2
....3....3.....3.....3.....3.....3.....3.....3.....3.....3.....3
....4....4.....4.....4.....4.....4.....4.....4.....4.....4.....4
....6....7.....7.....7.....7.....7.....7.....7.....7.....7.....7
....9...11....11....11....11....11....11....11....11....11....11
...13...17....18....18....18....18....18....18....18....18....18
...19...27....29....29....29....29....29....29....29....29....29
...28...42....46....47....47....47....47....47....47....47....47
...41...66....74....76....76....76....76....76....76....76....76
...60..104...118...122...123...123...123...123...123...123...123
...88..163...189...197...199...199...199...199...199...199...199
..129..256...303...317...321...322...322...322...322...322...322
..189..402...485...511...519...521...521...521...521...521...521
..277..631...777...824...838...842...843...843...843...843...843
..406..991..1244..1328..1354..1362..1364..1364..1364..1364..1364
..595.1556..1992..2141..2188..2202..2206..2207..2207..2207..2207
..872.2443..3190..3451..3535..3561..3569..3571..3571..3571..3571
.1278.3836..5108..5563..5712..5759..5773..5777..5778..5778..5778
.1873.6023..8180..8967..9229..9313..9339..9347..9349..9349..9349
.2745.9457.13099.14454.14912.15061.15108.15122.15126.15127.15127
Empirical: for n<=2k+1, T(n,k)=A080023(n)=A169985(n), which is A000032(n) for n>=2. - Danny Rorabaugh, Mar 13 2015

			Some solutions for n=6 k=4, one extended zero followed by filtered positions
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..0....0....0....0....0....0....0....0....0....0....0....0....0....0....0....0
..0....0....1....0....1....0....1....0....1....0....0....1....0....1....0....0
..0....0....0....1....0....0....0....1....0....0....0....0....0....0....1....1
..0....1....1....0....0....1....0....0....1....0....0....0....1....0....0....0
..0....0....0....0....0....0....0....0....0....1....0....0....0....1....1....0
..0....1....0....0....1....0....0....1....1....0....1....0....0....0....0....0
..1....0....0....1....0....1....1....0....0....0....0....0....0....0....0....0

R. H. Hardin, Table of n, a(n) for n = 1..9501

Column 1 is A000930(n+2).
Column 2 is A222122.
Columns 3 to 7 are A222329 to A222333.
Cf. A000032, A080023, A169985.

Empirical for column k:
k=1: a(n) = a(n-1)+a(n-3)
k=2: a(n) = a(n-1)+a(n-3)+a(n-5)
k=3: a(n) = a(n-1)+a(n-3)+a(n-5)+a(n-7)
k=4: a(n) = a(n-1)+a(n-3)+a(n-5)+a(n-7)+a(n-9)
k=5: a(n) = a(n-1)+a(n-3)+a(n-5)+a(n-7)+a(n-9)+a(n-11)
k=6: a(n) = a(n-1)+a(n-3)+a(n-5)+a(n-7)+a(n-9)+a(n-11)+a(n-13)
k=7: a(n) = a(n-1)+a(n-3)+a(n-5)+a(n-7)+a(n-9)+a(n-11)+a(n-13)+a(n-15)

A163733 Number of n X 2 binary arrays with all 1's connected, all corners 1, and no 1 having more than two 1's adjacent.

Original entry on oeis.org

1, 1, 2, 2, 4, 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, 754, 1220, 1974, 3194, 5168, 8362, 13530, 21892, 35422, 57314, 92736, 150050, 242786, 392836, 635622, 1028458, 1664080, 2692538, 4356618, 7049156, 11405774, 18454930, 29860704, 48315634, 78176338

Offset: 1

R. H. Hardin, Aug 03 2009

nonn

Same recurrence for A163695.
Same recurrence for A163714.
Appears to coincide with diagonal sums of A072405. - Paul Barry, Aug 10 2009
From Gary W. Adamson, Sep 15 2016: (Start)
Let the sequence prefaced with a 1: (1, 1, 1, 2, 2, 4, 6, ...) equate to r(x). Then (r(x) * r(x^2) * r(x^4) * r(x^8) * ...) = the Fibonacci sequence, (1, 1, 2, 3, 5, ...). Let M = the following production matrix:
  1, 0, 0, 0, 0, ...
  1, 0, 0, 0, 0, ...
  1, 1, 0, 0, 0, ...
  2, 1, 0, 0, 0, ...
  2, 1, 1, 0, 0, ...
  4, 2, 1, 0, 0, ...
  6, 2, 1, 1, 0, ...
  ...
Limit of the matrix power M^k as k->infinity results in a single column vector equal to the Fibonacci sequence. (End)
Apparently a(n) = A128588(n-2) for n > 3. - Georg Fischer, Oct 14 2018

			All solutions for n=8:
   1 1   1 1   1 1   1 1   1 1   1 1   1 1   1 1   1 1   1 1
   0 1   1 0   1 0   1 0   1 0   1 0   0 1   0 1   0 1   0 1
   0 1   1 0   1 0   1 0   1 1   1 0   0 1   0 1   1 1   0 1
   0 1   1 0   1 0   1 1   0 1   1 0   0 1   0 1   1 0   1 1
   0 1   1 0   1 1   0 1   0 1   1 0   0 1   1 1   1 0   1 0
   0 1   1 0   0 1   0 1   0 1   1 1   1 1   1 0   1 0   1 0
   0 1   1 0   0 1   0 1   0 1   0 1   1 0   1 0   1 0   1 0
   1 1   1 1   1 1   1 1   1 1   1 1   1 1   1 1   1 1   1 1
------
   1 1   1 1   1 1   1 1   1 1   1 1
   0 1   0 1   0 1   1 0   1 0   1 0
   1 1   1 1   0 1   1 0   1 1   1 1
   1 0   1 0   1 1   1 1   0 1   0 1
   1 1   1 0   1 0   0 1   0 1   1 1
   0 1   1 1   1 1   1 1   1 1   1 0
   0 1   0 1   0 1   1 0   1 0   1 0
   1 1   1 1   1 1   1 1   1 1   1 1

R. H. Hardin, Table of n, a(n) for n=1..100

Cf. A118658, A055389, A006355, A006355, A169985, A000045.

Cf. A128588. - Georg Fischer, Oct 14 2018

Join[{1, 1}, Table[2*Fibonacci[n], {n, 70}]] (* Vladimir Joseph Stephan Orlovsky, Feb 10 2012 *)
Table[Round[GoldenRatio^(k-1)] - Round[GoldenRatio^(k-1)/Sqrt[5]], {k, 1, 70}] (* Federico Provvedi, Mar 26 2013 *)

Empirical: a(n) = a(n-1) + a(n-2) for n >= 5.
G.f.: (1-x^3)/(1-x-x^2) (conjecture). - Paul Barry, Aug 10 2009
a(n) = round(phi^(k-1)) - round(phi^(k-1)/sqrt(5)), phi = (1 + sqrt(5))/2 (conjecture). - Federico Provvedi, Mar 26 2013
G.f.: 1 + 2*x - x*Q(0), where Q(k) = 1 + x^2 - (2*k+1)*x + x*(2*k-1 - x)/Q(k+1); (conjecture), (continued fraction). - Sergei N. Gladkovskii, Oct 05 2013
G.f.: If prefaced with a 1, (1, 1, 1, 2, 2, 4, ...): (1 - x^2 - x^4)/(1 - x - x^2); where the modified sequence satisfies A(x)/A(x^2), A(x) is the Fibonacci sequence. - Gary W. Adamson, Sep 15 2016

A306357 Number of nonempty subsets of {1, ..., n} containing no three cyclically successive elements.

Original entry on oeis.org

0, 1, 3, 6, 10, 20, 38, 70, 130, 240, 442, 814, 1498, 2756, 5070, 9326, 17154, 31552, 58034, 106742, 196330, 361108, 664182, 1221622, 2246914, 4132720, 7601258, 13980894, 25714874, 47297028, 86992798, 160004702, 294294530, 541292032, 995591266, 1831177830

Offset: 0

Gus Wiseman, Feb 10 2019

nonn

Cyclically successive means 1 is a successor of n.

Set partitions using these subsets are counted by A323949.

			The a(1) = 1 through a(5) = 20 stable subsets:
  {1}  {1}    {1}    {1}    {1}
       {2}    {2}    {2}    {2}
       {1,2}  {3}    {3}    {3}
              {1,2}  {4}    {4}
              {1,3}  {1,2}  {5}
              {2,3}  {1,3}  {1,2}
                     {1,4}  {1,3}
                     {2,3}  {1,4}
                     {2,4}  {1,5}
                     {3,4}  {2,3}
                            {2,4}
                            {2,5}
                            {3,4}
                            {3,5}
                            {4,5}
                            {1,2,4}
                            {1,3,4}
                            {1,3,5}
                            {2,3,5}
                            {2,4,5}

Cf. A000126, A000296, A001610, A001644, A169985, A306357, A323949, A323952, A323955.

stabsubs[g_]:=Select[Rest[Subsets[Union@@g]],Select[g,Function[ed,UnsameQ@@ed&&Complement[ed,#]=={}]]=={}&];
Table[Length[stabsubs[Partition[Range[n],3,1,1]]],{n,15}]

For n >= 3 we have a(n) = A001644(n) - 1.
From Chai Wah Wu, Jan 06 2020: (Start)
a(n) = 2*a(n-1) - a(n-4) for n > 6.
G.f.: x*(x^5 + x^4 - 2*x^3 + x + 1)/(x^4 - 2*x + 1). (End)

A324014 Number of self-complementary set partitions of {1, ..., n} with no cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 9, 16, 43, 89, 250, 571, 1639

Offset: 0

Gus Wiseman, Feb 12 2019

The complement of a set partition pi of {1, ..., n} is defined as n + 1 - pi (elementwise) on page 3 of Callan. For example, the complement of {{1,5},{2},{3,6},{4}} is {{1,4},{2,6},{3},{5}}.

			The  a(3) = 1 through a(6) = 9 self-complementary set partitions with no cyclical adjacencies:
  {{1}{2}{3}}  {{13}{24}}      {{14}{25}{3}}      {{135}{246}}
               {{1}{2}{3}{4}}  {{1}{24}{3}{5}}    {{13}{25}{46}}
                               {{1}{2}{3}{4}{5}}  {{14}{25}{36}}
                                                  {{1}{24}{35}{6}}
                                                  {{13}{2}{46}{5}}
                                                  {{14}{2}{36}{5}}
                                                  {{15}{26}{3}{4}}
                                                  {{1}{25}{3}{4}{6}}
                                                  {{1}{2}{3}{4}{5}{6}}

David Callan, On conjugates for set partitions and integer compositions [math.CO].

Cf. A000110, A000296, A001610, A080107 (self-complementary), A169985, A324012 (self-conjugate), A324015.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
cmp[stn_]:=Union[Sort[Max@@Join@@stn+1-#]&/@stn];
Table[Select[sps[Range[n]],And[cmp[#]==Sort[#],Total[If[First[#]==1&&Last[#]==n,1,0]+Count[Subtract@@@Partition[#,2,1],-1]&/@#]==0]&]//Length,{n,0,10}]

A324015 Number of nonempty subsets of {1, ..., n} containing no two cyclically successive elements.

Original entry on oeis.org

0, 1, 2, 3, 6, 10, 17, 28, 46, 75, 122, 198, 321, 520, 842, 1363, 2206, 3570, 5777, 9348, 15126, 24475, 39602, 64078, 103681, 167760, 271442, 439203, 710646, 1149850, 1860497, 3010348, 4870846, 7881195, 12752042, 20633238, 33385281, 54018520, 87403802

Offset: 0

Gus Wiseman, Feb 12 2019

nonn

Cyclically successive means 1 succeeds n.

After a(1) = 1, same as A001610 shifted once to the right. Also, a(n) = A169985(n) - 1.

			The a(6) = 17 stable subsets:
  {1}, {2}, {3}, {4}, {5}, {6},
  {1,3}, {1,4}, {1,5}, {2,4}, {2,5}, {2,6}, {3,5}, {3,6}, {4,6},
  {1,3,5}, {2,4,6}.

Cf. A000045, A000126, A000296, A001610, A001644, A169985, A306357, A324014.

stabsubs[g_]:=Select[Rest[Subsets[Union@@g]],Select[g,Function[ed,UnsameQ@@ed&&Complement[ed,#]=={}]]=={}&];
Table[Length[stabsubs[Partition[Range[n],2,1,1]]],{n,0,10}]

For n <= 3, a(n) = n. Otherwise, a(n) = a(n - 1) + a(n - 2) + 1.

A062724 a(n) = floor(tau^n) + 1, where tau = (1 + sqrt(5))/2.

Original entry on oeis.org

2, 2, 3, 5, 7, 12, 18, 30, 47, 77, 123, 200, 322, 522, 843, 1365, 2207, 3572, 5778, 9350, 15127, 24477, 39603, 64080, 103682, 167762, 271443, 439205, 710647, 1149852, 1860498, 3010350, 4870847, 7881197, 12752043, 20633240, 33385282

Offset: 0

Jason Earls, Jul 15 2001

nonn

Apart from the first term, this sequence also gives the ceiling of the powers of the golden ratio (cf. A169986). - Mohammad K. Azarian, Apr 14 2008

Harry J. Smith, Table of n, a(n) for n = 0..400

Equals A014217 + 1.

Cf. A169985, A169986.

Floor[GoldenRatio^Range[0,40]]+1 (* Harvey P. Dale, Dec 18 2019 *)

j=[]; for(n=0,60,t=(1+sqrt(5))/2; j=concat(j,floor((t^n))+1)); j

{ default(realprecision, 200); t=(1 + sqrt(5))/2; p=1; for (n=0, 400, if (n, p*=t); write("b062724.txt", n, " ", p\1 + 1) ) } \\ Harry J. Smith, Aug 09 2009

a(n) = 3*Fibonacci(n-1) + Fibonacci(n-2) + (n mod 2), n > 0. - Gary Detlefs, Dec 29 2010

A205579 a(n) = round(r^n) where r is the smallest Pisot number (real root r=1.3247179.. of x^3-x-1).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 7, 9, 13, 17, 22, 29, 39, 51, 68, 90, 119, 158, 209, 277, 367, 486, 644, 853, 1130, 1497, 1983, 2627, 3480, 4610, 6107, 8090, 10717, 14197, 18807, 24914, 33004, 43721, 57918, 76725, 101639, 134643, 178364, 236282, 313007, 414646, 549289, 727653, 963935, 1276942, 1691588, 2240877, 2968530, 3932465

Offset: 0

Joerg Arndt, Jan 29 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Pisot Number.
Index entries for linear recurrences with constant coefficients, signature (0,1,1).

Cf. A112639 (definition using floor() instead of round()).
Cf. A060006 (decimal expansion of r=1.32471795724475...).
Cf. A051016, A051017, A169985.

CoefficientList[Series[(1+x+x^2+x^9+x^10-x^12)/(1-x^2-x^3),{x,0,100}],x] (* Vincenzo Librandi, Aug 19 2012 *)
r = Root[x^3-x-1, 1]; Table[Round[r^i], {i,0,100 }] (* Jwalin Bhatt, Mar 27 2025 *)

default(realprecision, 110);
default(format, "g.15");
r=real(polroots(x^3-x-1)[1])
v=vector(66, n, round(r^(n-1)) )

Vec((1+x+x^2+x^9+x^10-x^12)/(1-x^2-x^3)+O(x^66))

G.f.: (1+x+x^2+x^9+x^10-x^12)/(1-x^2-x^3).
From Jwalin Bhatt, Mar 26 2025: (Start)
a(n) = round(((1/2+sqrt(23/108))^(1/3) + (1/2-sqrt(23/108))^(1/3))^n).
a(n) = a(n-2) + a(n-3) for n>=13. (End)

A323949 Number of set partitions of {1, ..., n} with no block containing three distinct cyclically successive vertices.

Original entry on oeis.org

1, 1, 2, 4, 10, 36, 145, 631, 3015, 15563, 86144, 508311, 3180930, 21018999, 146111543, 1065040886, 8117566366, 64531949885, 533880211566, 4587373155544, 40865048111424, 376788283806743, 3590485953393739, 35312436594162173, 357995171351223109, 3736806713651177702

Offset: 0

Gus Wiseman, Feb 10 2019

nonn

Cyclically successive means 1 is a successor of n.

			The a(1) = 1 through a(4) = 10 set partitions:
  {{1}}  {{1,2}}    {{1},{2,3}}    {{1,2},{3,4}}
         {{1},{2}}  {{1,2},{3}}    {{1,3},{2,4}}
                    {{1,3},{2}}    {{1,4},{2,3}}
                    {{1},{2},{3}}  {{1},{2},{3,4}}
                                   {{1},{2,3},{4}}
                                   {{1,2},{3},{4}}
                                   {{1},{2,4},{3}}
                                   {{1,3},{2},{4}}
                                   {{1,4},{2},{3}}
                                   {{1},{2},{3},{4}}

Alois P. Heinz, Table of n, a(n) for n = 0..250

Cf. A000085, A000110, A000126, A000296, A000325, A001610, A001644, A078012, A169985.

Cf. A306351, A306357, A323950, A323951, A323953, A323954, A323955, A323956.

spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],Select[Partition[Range[n],3,1,1],Function[ed,UnsameQ@@ed&&Complement[ed,#]=={}]]=={}&],Range[n]]],{n,8}]

a(12)-a(25) from Alois P. Heinz, Feb 10 2019

cp's OEIS Frontend

A306417 Number of self-conjugate set partitions of {1, ..., n}.

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A324012 Number of self-complementary set partitions of {1, ..., n} with no singletons or cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

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A222334 T(n,k)=Number of binary arrays indicating the locations of trailing edge maxima of a random length-n 0..k array extended with zeros and convolved with 1,1.

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A163733 Number of n X 2 binary arrays with all 1's connected, all corners 1, and no 1 having more than two 1's adjacent.

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A306357 Number of nonempty subsets of {1, ..., n} containing no three cyclically successive elements.

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A324014 Number of self-complementary set partitions of {1, ..., n} with no cyclical adjacencies (successive elements in the same block, where 1 is a successor of n).

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A324015 Number of nonempty subsets of {1, ..., n} containing no two cyclically successive elements.

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A062724 a(n) = floor(tau^n) + 1, where tau = (1 + sqrt(5))/2.

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