A009964 -id:A009964 - cp's OEIS frontend

A087752 Powers of 49.

1, 49, 2401, 117649, 5764801, 282475249, 13841287201, 678223072849, 33232930569601, 1628413597910449, 79792266297612001, 3909821048582988049, 191581231380566414401, 9387480337647754305649, 459986536544739960976801, 22539340290692258087863249, 1104427674243920646305299201

Offset: 0

Douglas Winston (douglas.winston(AT)srupc.com), Oct 02 2003

Same as Pisot sequences E(1, 49), L(1, 49), P(1, 49), T(1, 49). Essentially same as Pisot sequences E(49, 2401), L(49, 2401), P(49, 2401), T(49, 2401). See A008776 for definitions of Pisot sequences.

The compositions of n in which each natural number is colored by one of p different colors are called p-colored compositions of n. For n >= 1, a(n) equals the number of 49-colored compositions of n such that no adjacent parts have the same color. - Milan Janjic, Nov 17 2011

T. D. Noe, Table of n, a(n) for n = 0..100
Tanya Khovanova, Recursive Sequences.
Index entries for linear recurrences with constant coefficients, signature (49).

Bisection of A000420.
Cf. A001018 (powers of 8), ..., A001029 (powers of 19), A009964 (powers of 20), ..., A009992 (powers of 48).
Cf. A005843, A008776.

[49^n: n in [0..20]]; // Vincenzo Librandi, Nov 21 2010

49^Range[0,20] (* or *) Join[{1},NestList[49#&,49,20]] (* Harvey P. Dale, May 10 2019 *)

makelist(49^n,n,0,20); /* Martin Ettl, Nov 12 2012 */

PARI
```
a(n)=49^n \\ M. F. Hasler, Apr 19 2015
```

G.f.: 1/(1-49*x). - Philippe Deléham, Nov 24 2008
From Vincenzo Librandi, Nov 21 2010: (Start)
a(n) = 49^n.
a(n) = 49*a(n-1), a(0)=1. (End)
From Elmo R. Oliveira, Jul 08 2025: (Start)
E.g.f.: exp(49*x).
a(n) = A000420(A005843(n)). (End)

Edited by M. F. Hasler, Apr 19 2015

A223480 T(n,k)=Rolling icosahedron face footprints: number of nXk 0..19 arrays starting with 0 where 0..19 label faces of an icosahedron and every array movement to a horizontal or antidiagonal neighbor moves across an icosahedral edge.

Original entry on oeis.org

1, 3, 20, 9, 27, 400, 27, 135, 243, 8000, 81, 675, 2025, 2187, 160000, 243, 3375, 16875, 30375, 19683, 3200000, 729, 16875, 147825, 421875, 455625, 177147, 64000000, 2187, 84375, 1296675, 6526575, 10546875, 6834375, 1594323, 1280000000, 6561, 421875

Offset: 1

R. H. Hardin Mar 20 2013

Table starts
..........1........3..........9...........27.............81..............243
.........20.......27........135..........675...........3375............16875
........400......243.......2025........16875.........147825..........1296675
.......8000.....2187......30375.......421875........6526575........101331675
.....160000....19683.....455625.....10546875......288507825.......7939566675
....3200000...177147....6834375....263671875....12755926575.....622332801675
...64000000..1594323..102515625...6591796875...563999907825...48783753036675
.1280000000.14348907.1537734375.164794921875.24937217326575.3824122400271675

			Some solutions for n=3 k=4
..0..1..6.10....0..1..4..1....0..1..0..1....0..2..8.13....0..2..8..9
..6..1..6..1....6..1..4..3....4..1..0..5....0..2..8..2....8..9..8..9
..4..1..4..3....4.17..4.17....0..5..9..5....8..2..3..4....8..2..8..9
Face neighbors:
0 -> 1 2 5
1 -> 0 4 6
2 -> 0 3 8
3 -> 2 4 16
4 -> 3 1 17
5 -> 0 7 9
6 -> 1 7 10
7 -> 6 5 11
8 -> 2 9 13
9 -> 8 5 14
10 -> 6 12 17
11 -> 7 12 14
12 -> 11 10 19
13 -> 8 15 16
14 -> 9 11 15
15 -> 14 13 19
16 -> 3 13 18
17 -> 4 10 18
18 -> 16 17 19
19 -> 15 18 12

R. H. Hardin, Table of n, a(n) for n = 1..160

Column 1 is A009964(n-1)
Column 2 is A013708(n-1)
Column 3 is 9*15^(n-1)
Column 4 is 27*25^(n-1)
Row 1 is A000244(n-1)
Row 2 is 27*5^(n-2) for n>1

Empirical for column k:
k=1: a(n) = 20*a(n-1)
k=2: a(n) = 9*a(n-1)
k=3: a(n) = 15*a(n-1)
k=4: a(n) = 25*a(n-1)
k=5: a(n) = 51*a(n-1) -300*a(n-2)
k=6: a(n) = 101*a(n-1) -1900*a(n-2) +10000*a(n-3)
k=7: a(n) = 227*a(n-1) -14764*a(n-2) +411840*a(n-3) -5347200*a(n-4) +29600000*a(n-5) -48000000*a(n-6)
Empirical for row n:
n=1: a(n) = 3*a(n-1)
n=2: a(n) = 5*a(n-1) for n>2
n=3: a(n) = 9*a(n-1) -2*a(n-2) for n>4
n=4: a(n) = 17*a(n-1) -16*a(n-2) -76*a(n-3) +64*a(n-4) for n>7
n=5: a(n) = 33*a(n-1) -86*a(n-2) -1564*a(n-3) +7040*a(n-4) -6480*a(n-5) -5088*a(n-6) +5824*a(n-7) -512*a(n-8) for n>13
n=6: [order 20] for n>25

A359452 Number of vertices in the partite set of the n-Menger sponge graph that contains the corners.

Original entry on oeis.org

1, 8, 208, 3968, 80128, 1599488, 32002048, 639991808, 12800032768, 255999868928, 5120000524288, 102399997902848, 2048000008388608, 40959999966445568, 819200000134217728, 16383999999463129088, 327680000002147483648, 6553599999991410065408, 131072000000034359738368

Offset: 0

Allan Bickle, Jan 02 2023

This sequence and the sequence counting the non-corner vertices (A359453) alternate as to which is larger.

			The level 1 Menger sponge graph can be formed by subdividing every edge of a cube graph.  This produces a graph with 8 corner vertices and 12 non-corner vertices, so a(1) = 8.

Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Eric Weisstein's World of Mathematics, Menger Sponge
Wikipedia, Menger sponge
Index entries for linear recurrences with constant coefficients, signature (16,80).

Cf. A009964 (number of vertices), A291066 (number of edges).
Cf. A359453 (number of non-corner vertices).
Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.
Cf. A262710.

A359452[n_]:=(20^n+(-4)^n)/2;Array[A359452,25,0] (* Paolo Xausa, Nov 29 2023 *)

a(n) = (20^n + (-4)^n)/2 \\ Andrew Howroyd, Jan 02 2023

def A359452(n): return (10**n<Chai Wah Wu, Feb 13 2023

a(n) = (20^n + (-4)^n)/2.
a(n) = (A009964(n) + A262710(n))/2.
a(n) = 20^n - A359453(n).
From Stefano Spezia, Jan 02 2023: (Start)
O.g.f.: (1 - 8*x)/((1 - 20*x)*(1 + 4*x)).
E.g.f.: exp(8*x)*cosh(12*x). (End)

A359453 Number of vertices in the partite set of the n-Menger sponge graph that do not contain the corners.

Original entry on oeis.org

0, 12, 192, 4032, 79872, 1600512, 31997952, 640008192, 12799967232, 256000131072, 5119999475712, 102400002097152, 2047999991611392, 40960000033554432, 819199999865782272, 16384000000536870912, 327679999997852516352, 6553600000008589934592, 131071999999965640261632

Offset: 0

Allan Bickle, Jan 02 2023

This sequence and the sequence counting the corner vertices (A359452) alternate as to which is larger.

			The level 1 Menger sponge graph can be formed by subdividing every edge of a cube graph.  This produces a graph with 8 corner vertices and 12 non-corner vertices, so a(1) = 12.

Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Eric Weisstein's World of Mathematics, Menger Sponge
Wikipedia, Menger sponge
Index entries for linear recurrences with constant coefficients, signature (16,80).

Cf. A009964 (number of vertices), A291066 (number of edges).
Cf. A359452 (number of corner vertices).
Cf. A291066, A083233, and A332705 on the surface area of the n-Menger sponge graph.

A359453[n_]:=(20^n-(-4)^n)/2;Array[A359453,25,0] (* Paolo Xausa, Nov 30 2023 *)

a(n) = (20^n - (-4)^n)/2 \\ Andrew Howroyd, Jan 02 2023

def A359453(n): return (10**n<Chai Wah Wu, Feb 13 2023

a(n) = (20^n - (-4)^n)/2.
a(n) = (A009964(n) - A262710(n))/2.
a(n) = 20^n - A359452(n).
From Stefano Spezia, Jan 02 2023: (Start)
O.g.f.: 12*x/((1 - 20*x)*(1 + 4*x)).
E.g.f.: (cosh(8*x) + sinh(8*x))*sinh(12*x). (End)

A365606 Number of degree 2 vertices in the n-Sierpinski carpet graph.

Original entry on oeis.org

8, 20, 84, 500, 3540, 26996, 212052, 1684724, 13442772, 107437172, 859182420, 6872514548, 54977282004, 439809752948, 3518452514388, 28147543587572, 225180119118036, 1801440264196724, 14411520047331156, 115292154179921396, 922337214843187668, 7378697662956950900, 59029581136289955924

Offset: 1

Allan Bickle, Sep 12 2023

The level 0 Sierpinski carpet graph is a single vertex. The level n Sierpinski carpet graph is formed from 8 copies of level n-1 by joining boundary vertices between adjacent copies.

			The level 1 Sierpinski carpet graph is an 8-cycle, which has 8 degree 2 vertices and 0 degree 3 or 4 vertices.  Thus a(1) = 8.

Paolo Xausa, Table of n, a(n) for n = 1..1000
Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Eric Weisstein's World of Mathematics, Sierpiński Carpet Graph
Index entries for linear recurrences with constant coefficients, signature (12,-35,24).

Cf. A001018 (order), A271939 (size).
Cf. A365606 (degree 2), A365607 (degree 3), A365608 (degree 4).
Cf. A009964, A291066, A359452, A359453, A291066, A083233, A332705 (Menger sponge graph).

LinearRecurrence[{12,-35,24},{8,20,84},30] (* Paolo Xausa, Oct 16 2023 *)

def A365606(n): return ((1<<3*n-1)+(3**(n-1)<<4))//5+4 # Chai Wah Wu, Nov 27 2023

a(n) = (1/10)*8^n + (16/15)*3^n + 4.
a(n) = 8*a(n-1) - 16*3^(n-2) - 28.
a(n) = 8^n - A365607(n) - A365608(n).
2*a(n) = 2*A271939(n) - 3*A365607(n) - 4*A365608(n).
G.f.: 4*x*(2 - 19*x + 31*x^2)/((1 - x)*(1 - 3*x)*(1 - 8*x)). - Stefano Spezia, Sep 12 2023

A365607 Number of degree 3 vertices in the n-Sierpinski carpet graph.

Original entry on oeis.org

0, 40, 328, 2536, 19912, 158056, 1260616, 10073320, 80551624, 644308072, 5154149704, 41232252904, 329855188936, 2638833008488, 21110638558792, 168885031942888, 1351080025960648, 10808639518937704, 86469114085259080, 691752906483344872, 5534023233270575560, 44272185810376054120

Offset: 1

Allan Bickle, Sep 12 2023

The level 0 Sierpinski carpet graph is a single vertex. The level n Sierpinski carpet graph is formed from 8 copies of level n-1 by joining boundary vertices between adjacent copies.

			The level 1 Sierpinski carpet graph is an 8-cycle, which has 8 degree 2 vertices and 0 degree 3 or 4 vertices.  Thus a(1) = 0.

Paolo Xausa, Table of n, a(n) for n = 1..1000
Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Eric Weisstein's World of Mathematics, Sierpiński Carpet Graph
Index entries for linear recurrences with constant coefficients, signature (12,-35,24).

Cf. A001018 (order), A271939 (size).
Cf. A365606 (degree 2), A365607 (degree 3), A365608 (degree 4).
Cf. A009964, A291066, A359452, A359453, A291066, A083233, A332705 (Menger sponge graph).

LinearRecurrence[{12,-35,24},{0,40,328},30] (* Paolo Xausa, Oct 16 2023 *)

def A365607(n): return ((3<<3*n)+(3**(n-1)<<4))//5-8 # Chai Wah Wu, Nov 27 2023

a(n) = (3/5)*8^n + (16/15)*3^n - 8.
a(n) = 8*a(n-1) - 16*3^(n-2) + 56.
a(n) = 8^n - A365606(n) - A365608(n).
3*a(n) = 2*A271939(n) - 2*A365606(n) - 4*A365608(n).
G.f.: 8*x^2*(5 - 19*x)/((1 - x)*(1 - 3*x)*(1 - 8*x)). - Stefano Spezia, Sep 12 2023

A365608 Number of degree 4 vertices in the n-Sierpinski carpet graph.

Original entry on oeis.org

0, 4, 100, 1060, 9316, 77092, 624484, 5019172, 40223332, 321996580, 2576602468, 20614709284, 164923342948, 1319403749668, 10555281015652, 84442401180196, 675539668606564, 5404318726347556, 43234553943265636, 345876443943580708, 2767011588741012580, 22136092821505201444, 177088742906772914020

Offset: 1

Allan Bickle, Sep 12 2023

The level 0 Sierpinski carpet graph is a single vertex. The level n Sierpinski carpet graph is formed from 8 copies of level n-1 by joining boundary vertices between adjacent copies.

			The level 1 Sierpinski carpet graph is an 8-cycle, which has 8 degree 2 vertices and 0 degree 3 or 4 vertices.  Thus a(1) = 0.

Paolo Xausa, Table of n, a(n) for n = 1..1000
Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Eric Weisstein's World of Mathematics, Sierpiński Carpet Graph
Index entries for linear recurrences with constant coefficients, signature (12,-35,24).

Cf. A001018 (order), A271939 (size).
Cf. A365606 (degree 2), A365607 (degree 3), A365608 (degree 4).
Cf. A009964, A291066, A359452, A359453, A291066, A083233, A332705 (Menger sponge graph).

LinearRecurrence[{12,-35,24},{0,4,100},30] (* Paolo Xausa, Oct 16 2023 *)

def A365608(n): return ((3<<3*n-1)-(3**(n-1)<<5))//5+4 # Chai Wah Wu, Nov 27 2023

a(n) = (3/10)*8^n - (32/15)*3^n + 4.
a(n) = 8*a(n-1) + 32*3^(n-2) - 28.
a(n) = 8^n - A365606(n) - A365607(n).
4*a(n) = 2*A271939(n) - 2*A365606(n) - 3*A365607(n).
G.f.: 4*x^2*(1 + 13*x)/((1 - x)*(1 - 3*x)*(1 - 8*x)). - Stefano Spezia, Sep 12 2023

A367700 Number of degree 2 vertices in the n-Menger sponge graph.

Original entry on oeis.org

12, 72, 744, 11256, 201960, 3871416, 76138536, 1512609912, 30171384168, 602782587960, 12050495247528, 240968665611768, 4819043435788776, 96378229818994104, 1927543485550004520, 38550700825394191224, 771012665426135994984, 15420242499878035355448, 308404763528431125030312

Offset: 1

Allan Bickle, Nov 27 2023

The level 0 Menger sponge graph is a single vertex. The level n Menger sponge graph is formed from 20 copies of level n-1 in the shape of a cube with middle faces removed by joining boundary vertices between adjacent copies.

			The level 1 Menger sponge graph is a cube with each edge subdivided, which has 12 degree 2 vertices and 8 degree 3 vertices.  Thus a(1) = 12.

Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Eric Weisstein's World of Mathematics, Menger Sponge Graph.
Index entries for linear recurrences with constant coefficients, signature (31,-244,480).

Cf. A009964 (number of vertices), A291066 (number of edges).
Cf. A359452, A359453 (numbers of corner and non-corner vertices).
Cf. A083233, A332705 (surface area).
Cf. A367701, A367702, A367706, A367707 (degrees 2 through 6).
Cf. A001018, A271939, A365602, A365606, A365607, A365608 (Sierpinski carpet graphs).

LinearRecurrence[{31,-244,480}, {12, 72, 744}, 25] (* Paolo Xausa, Nov 28 2023 *)

def A367700(n): return (5*20**n+(34<<3*n)+216*3**n)//85 # Chai Wah Wu, Nov 27 2023

a(n) = (1/17)*20^n + (2/5)*8^n + (216/85)*3^n.
a(n) = 20*a(n-1) - (3/5)*8^n - (72/5)*3^n.
a(n) = 20^n - A367701(n) - A367702(n) - A367706(n) - A367707(n).
2*a(n) = 2*A291066(n) - 3*A367701(n) - 4*A365602(n) - 5*A367706(n) - 6*A367707(n).
G.f.: 12*x*(1 - 25*x + 120*x^2)/((1 - 3*x)*(1 - 8*x)*(1 - 20*x)). - Stefano Spezia, Nov 27 2023

A367701 Number of degree 3 vertices in the n-Menger sponge graph.

Original entry on oeis.org

8, 152, 2744, 49688, 941624, 18381464, 363917240, 7248334616, 144725667128, 2892582307736, 57836189374136, 1156600107729944, 23131012640050232, 462612336455034008, 9252183397644168632, 185043161299165038872, 3700859172747355380536, 74017151029040948253080

Offset: 1

Allan Bickle, Nov 27 2023

The level 0 Menger sponge graph is a single vertex. The level n Menger sponge graph is formed from 20 copies of level n-1 in the shape of a cube with middle faces removed by joining boundary vertices between adjacent copies.

			The level 1 Menger sponge graph is a cube with each edge subdivided, which has 12 degree 2 vertices and 8 degree 3 vertices.  Thus a(1) = 8.

Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Eric Weisstein's World of Mathematics, Menger Sponge Graph.
Index entries for linear recurrences with constant coefficients, signature (32,-275,724,-480).

Cf. A009964 (number of vertices), A291066 (number of edges).
Cf. A359452, A359453 (numbers of corner and non-corner vertices).
Cf. A291066, A083233, A332705 (surface area).
Cf. A367700, A367701, A367702, A367706, A367707 (degrees 2 through 6).
Cf. A001018, A271939, A365606, A365607, A365608 (Sierpinski carpet graphs).

LinearRecurrence[{32,-275,724,-480},{8,152,2744,49688},25] (* Paolo Xausa, Nov 28 2023 *)

def A367701(n): return ((3*5**n<<(n<<1)+3)+(51<<(3*n+1))-(3**(n+3)<<4))//85+8 # Chai Wah Wu, Nov 28 2023

a(n) = (24/85)*20^n + (6/5)*8^n - (432/85)*3^n + 8.
a(n) = 20*a(n-1) - (9/5)*8^n + (144/5)*3^n - 152.
a(n) = 20^n - A367700(n) - A367702(n) - A367706(n) - A367707(n).
3*a(n) = 2*A291066(n) - 2*A367700(n) - 4*A365602(n) - 5*A367706(n) - 6*A367707(n).
G.f.: 8*x*(1 - 13*x + 10*x^2 - 264*x^3)/((1 - x)*(1 - 3*x)*(1 - 8*x)*(1 - 20*x)). - Stefano Spezia, Nov 27 2023

A367702 Number of degree 4 vertices in the n-Menger sponge graph.

Original entry on oeis.org

0, 144, 2784, 57552, 1180320, 23889936, 480221280, 9624275280, 192645717024, 3854200280208, 77094305873376, 1541968557881808, 30840030795738528, 616805893363960080, 12336160087905835872, 246723539526229152336, 4934473492678780614432, 98689491470837087102352

Offset: 1

Allan Bickle, Nov 27 2023

The level 0 Menger sponge graph is a single vertex. The level n Menger sponge graph is formed from 20 copies of level n-1 in the shape of a cube with middle faces removed by joining boundary vertices between adjacent copies.

			The level 1 Menger sponge graph is a cube with each edge subdivided, which has 12 degree 2 vertices and 8 degree 3 vertices.  Thus a(1) = 0.

Allan Bickle, Degrees of Menger and Sierpinski Graphs, Congr. Num. 227 (2016) 197-208.
Allan Bickle, MegaMenger Graphs, The College Mathematics Journal, 49 1 (2018) 20-26.
Index entries for linear recurrences with constant coefficients, signature (32,-275,724,-480).

Cf. A009964 (number of vertices), A291066 (number of edges).
Cf. A359452, A359453 (numbers of corner and non-corner vertices).
Cf. A291066, A083233, A332705 (surface area).
Cf. A367700, A367701, A367702, A367706, A367707 (degrees 2 through 6).
Cf. A001018, A271939, A365606, A365607, A365608 (Sierpinski carpet graphs).

LinearRecurrence[{32,-275,724,-480},{0,144,2784,57552},25] (* Paolo Xausa, Nov 29 2023 *)

def A367702(n): return ((5**n<<(n<<1)+5)-(17<<(3*n+2))+(3**(n+4)<<3))//85-24 # Chai Wah Wu, Nov 28 2023

a(n) = (32/85)*20^n - (4/5)*8^n + (648/85)*3^n - 24.
a(n) = 20*a(n-1) + (6/5)*8^n - (216/5)*3^n + 456.
a(n) = 20^n - A367700(n) - A367701(n) - A367706(n) - A367707(n).
4*a(n) = 2*A291066(n) - 2*A367700(n) - 3*A367701(n) - 5*A367706(n) - 6*A367707(n).
G.f.: 12*x^2*(7 - 224*x + 1865*x^2 - 4308*x^3)/(5*(1 - x)*(1 - 3*x)*(1 - 8*x)*(1 - 20*x)). - Stefano Spezia, Nov 28 2023

cp's OEIS Frontend

A087752 Powers of 49.

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A223480 T(n,k)=Rolling icosahedron face footprints: number of nXk 0..19 arrays starting with 0 where 0..19 label faces of an icosahedron and every array movement to a horizontal or antidiagonal neighbor moves across an icosahedral edge.

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A359452 Number of vertices in the partite set of the n-Menger sponge graph that contains the corners.

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A359453 Number of vertices in the partite set of the n-Menger sponge graph that do not contain the corners.

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A365606 Number of degree 2 vertices in the n-Sierpinski carpet graph.

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A365607 Number of degree 3 vertices in the n-Sierpinski carpet graph.

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A365608 Number of degree 4 vertices in the n-Sierpinski carpet graph.

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