A285395 -id:A285395 - cp's OEIS frontend

A285391 Start with a single cell at coordinates (0, 0), then iteratively subdivide the grid into 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 2; a(n) is the number of cells after n iterations.

1, 8, 60, 444, 3276, 24156, 178092, 1312956, 9679500, 71360028, 526086252, 3878455932, 28593068364, 210796144092, 1554048476460, 11456882559036, 84463361313804, 622687661115804, 4590628614276588, 33843405595099644, 249503106984577740, 1839407095720003932

Offset: 0

Peter Karpov, Apr 18 2017

Cell configuration converges to a fractal carpet with dimension 1.818...

Colin Barker, Table of n, a(n) for n = 0..1000
Peter Karpov, InvMem, Item 26
Peter Karpov, Illustration of cell configuration after 5 iterations
Index entries for linear recurrences with constant coefficients, signature (9,-12).

Cf. A285392, A285393, A285394, A285395, A285396, A285397, A285398, A285399, A285400.

[n le 2 select 8^(n-1) else 9*Self(n-1) - 12*Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 11 2021

Mathematica
```
LinearRecurrence[{9, -12}, {1, 8}, 16]
```

Vec((1 - x) / (1 - 9*x + 12*x^2) + O(x^30)) \\ Colin Barker, Apr 18 2017

[(2*sqrt(3))^(n-1)*( 2*sqrt(3)*chebyshev_U(n, 9/(4*sqrt(3))) - chebyshev_U(n-1, 9/(4*sqrt(3))) ) for n in (0..30)] # G. C. Greubel, Dec 11 2021

a(0) = 1, a(1) = 8, a(n) = 9*a(n-1) - 12*a(n-2).
G.f.: (1-x)/(1-9*x+12*x^2).
a(n) = (2^(-1-n)*((9-sqrt(33))^n*(-7+sqrt(33)) + (7+sqrt(33))*(9+sqrt(33))^n)) / sqrt(33). - Colin Barker, Apr 18 2017
a(n) = (2*sqrt(3))^(n-1)*( 2*sqrt(3)*ChebyshevU(n, 9/(4*sqrt(3))) - ChebyshevU(n-1, 9/(4*sqrt(3))) ). - G. C. Greubel, Dec 11 2021

More terms from Colin Barker, Apr 18 2017

A285392 Start with a single cell at coordinates (0, 0), then iteratively subdivide the grid into 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 0; a(n) is the number of cells after n iterations.

Original entry on oeis.org

1, 5, 36, 264, 1944, 14328, 105624, 778680, 5740632, 42321528, 312006168, 2300197176, 16957700568, 125016939000, 921660044184, 6794737129656, 50092713636696, 369297577174392, 2722565630929176, 20071519752269880, 147972890199278808, 1090897774766270712

Offset: 0

Peter Karpov, Apr 18 2017

Cell configuration converges to a fractal carpet with dimension 1.818...

Colin Barker, Table of n, a(n) for n = 0..1000
Peter Karpov, InvMem, Item 26
Peter Karpov, Illustration of cell configuration after 5 iterations
Index entries for linear recurrences with constant coefficients, signature (9,-12).

Cf. A285391, A285393, A285394, A285395, A285396, A285397, A285398, A285399, A285400.

I:=[5, 36]; [1] cat [n le 2 select I[n] else 9*Self(n-1) - 12*Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 11 2021

{1}~Join~LinearRecurrence[{9, -12}, {5, 36}, 16]

Vec((1 - x)*(1 - 3*x) / (1 - 9*x + 12*x^2) + O(x^30)) \\ Colin Barker, Apr 18 2017

[(1/4)*(bool(n==0) + (2*sqrt(3))^(n-1)*( 6*sqrt(3)*chebyshev_U(n, 9/(4*sqrt(3))) - 7*chebyshev_U(n-1, 9/(4*sqrt(3))) ) ) for n in (0..30)] # G. C. Greubel, Dec 11 2021

a(0) = 1, a(1) = 5, a(2) = 36, a(n) = 9*a(n-1) - 12*a(n-2).
G.f.: (1-4*x+3*x^2)/(1-9*x+12*x^2).
a(n) = (2^(-3-n)*((9-sqrt(33))^n*(-13+3*sqrt(33)) + (9+sqrt(33))^n*(13+3*sqrt(33)))) / sqrt(33) for n>0. - Colin Barker, Apr 18 2017
a(n) = (1/4)*([n=0] + (2*sqrt(3))^(n-1)*( 6*sqrt(3)*ChebyshevU(n, 9/(4*sqrt(3))) - 7*ChebyshevU(n-1, 9/(4*sqrt(3))) ) ). - G. C. Greubel, Dec 11 2021

More terms from Colin Barker, Apr 18 2017

A285393 Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 2 or 3; a(n) is the number of cells after n iterations.

Original entry on oeis.org

1, 20, 352, 6080, 104704, 1802240, 31019008, 533872640, 9188540416, 158144921600, 2721848492032, 46846013603840, 806271544459264, 13876822236200960, 238835410589974528, 4110620744461844480, 70748315180918112256, 1217656507884193710080, 20957211028999804813312

Offset: 0

Peter Karpov, Apr 19 2017

nonn

Cell configuration converges to a fractal sponge with dimension 2.590...

G. C. Greubel, Table of n, a(n) for n = 0..750
Peter Karpov, InvMem, Item 26
Peter Karpov, Illustration of initial terms (n = 1..4)
Index entries for linear recurrences with constant coefficients, signature (20,-48).

Cf. A285391, A285392, A285394, A285395, A285396, A285397, A285398, A285399, A285400.

[n le 2 select (20)^(n-1) else 20*Self(n-1) - 48*Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 11 2021

LinearRecurrence[{20, -48}, {1, 20}, 19]

[(4*sqrt(3))^n * chebyshev_U(n, 5/(2*sqrt(3))) for n in (0..30)] # G. C. Greubel, Dec 11 2021

a(0) = 1, a(1) = 20, a(n) = 20*a(n-1) - 48*a(n-2).
G.f.: 1/(1-20*x+48*x^2).
a(n) = ((13 - 5*sqrt(13))*(10 - 2*sqrt(13))^n + (2*(5 + sqrt(13)))^n*(13 + 5*sqrt(13)))/26.
a(n) = (4*sqrt(3))^n * ChebyshevU(n, 5/(2*sqrt(3))). - G. C. Greubel, Dec 11 2021

A285394 Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 0 or 1; a(n) is the number of cells after n iterations.

Original entry on oeis.org

1, 7, 116, 1984, 34112, 587008, 10102784, 173879296, 2992652288, 51506839552, 886489481216, 15257461325824, 262597731418112, 4519596484722688, 77787238586384384, 1338804140460998656, 23042295357073522688, 396583308399342518272, 6825635990847321276416

Offset: 0

Peter Karpov, Apr 19 2017

nonn

Cell configuration converges to a fractal with dimension 2.590...

G. C. Greubel, Table of n, a(n) for n = 0..750
Peter Karpov, InvMem, Item 26
Peter Karpov, Illustration of initial terms (n = 1..4)
Index entries for linear recurrences with constant coefficients, signature (20,-48).

Cf. A285391, A285392, A285393, A285395, A285396, A285397, A285398, A285399, A285400.

I:=[7,116]; [n le 2 select I[n] else 20*Self(n-1) - 48*Self(n-2): n in [1..31]]; // G. C. Greubel, Dec 10 2021

{1}~Join~LinearRecurrence[{20, -48}, {7, 116}, 18]
CoefficientList[Series[(1 - 13x + 24x^2)/(1 - 20x + 48x^2), {x, 0, 40}], x] (* Indranil Ghosh, Apr 19 2017 *)

[(1/2)*bool(n==0) + (4*sqrt(3))^(n-1)*(2*sqrt(3)*chebyshev_U(n, 5/(2*sqrt(3))) - 3*chebyshev_U(n-1, 5/(2*sqrt(3)))) for n in (0..30)] # G. C. Greubel, Dec 10 2021

a(0) = 1, a(1) = 7, a(2) = 116, a(n) = 20*a(n-1) - 48*a(n-2).
G.f.: (1-13*x+24*x^2)/(1-20*x+48*x^2).
a(n) = (3*(10-2*sqrt(13))^n*(13+sqrt(13)) + (2*(5+sqrt(13)))^n*(91+23*sqrt(13)))/(52*(5+sqrt(13))) for n > 0.
a(n) = (1/2)*[n=0] + (4*sqrt(3))^(n-1)*(2*sqrt(3)*ChebyshevU(n, 5/(2*sqrt(3))) - 3*ChebyshevU(n-1, 5/(2*sqrt(3)))). - G. C. Greubel, Dec 10 2021

A285396 Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 2; a(n) is the number of cells after n iterations.

Original entry on oeis.org

1, 21, 399, 7401, 136227, 2500437, 45845895, 840237393, 15396839067, 282119272221, 5169192919455, 94712719519353, 1735370171447763, 31796203000166949, 582583421696631159, 10674336158022192609, 195579614965832408523, 3583490696858688375405

Offset: 0

Peter Karpov, Apr 19 2017

Cell configuration converges to a fractal with dimension 2.647...

G. C. Greubel, Table of n, a(n) for n = 0..750
Peter Karpov, InvMem, Item 26
Peter Karpov, Illustration of initial terms (n = 1..4)
Index entries for linear recurrences with constant coefficients, signature (28,-195,324).

Cf. A285391, A285392, A285393, A285394, A285395, A285397, A285398, A285399, A285400.

I:=[1,21,399]; [n le 3 select I[n] else 28*Self(n-1) - 195*Self(n-2) + 324*Self(n-3) : n in [1..41]]; // G. C. Greubel, Dec 10 2021

LinearRecurrence[{28, -195, 324}, {1, 21, 399}, 20]

def A285396_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-7*x+6*x^2)/(1-28*x+195*x^2-324*x^3) ).list()
A285396_list(40) # G. C. Greubel, Dec 10 2021

a(0) = 1, a(1) = 21, a(2) = 399, a(n) = 28*a(n-1) - 195*a(n-2) + 324*a(n-3).

G.f.: (1-7*x+6*x^2)/(1-28*x+195*x^2-324*x^3).

A285397 Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 3; a(n) is the number of cells after n iterations.

Original entry on oeis.org

1, 26, 646, 15818, 385822, 9401330, 229023958, 5578844858, 135894050926, 3310204057250, 80632220390758, 1964094376340522, 47842741143064894, 1165385872796078546, 28387257791866411894, 691476036231391881242, 16843441238514542846350, 410283940250387099210114

Offset: 0

Peter Karpov, Apr 23 2017

Cell configuration converges to a fractal with dimension 2.906...

Colin Barker, Table of n, a(n) for n = 0..700
Peter Karpov, InvMem, Item 26
Index entries for linear recurrences with constant coefficients, signature (32,-195,216).

Cf. A007482, A026597, A285391, A285392, A285393, A285394, A285395, A285396, A285398, A285399, A285400.

I:=[1, 26, 646]; [n le 3 select I[n] else 32*Self(n-1) - 195*Self(n-2) + 216*Self(n-3) : n in [1..41]]; // G. C. Greubel, Dec 09 2021

LinearRecurrence[{32, -195, 216}, {1, 26, 646}, 18]

Vec((1 - 3*x)^2 / (1 - 32*x + 195*x^2 - 216*x^3) + O(x^20)) \\ Colin Barker, Apr 23 2017

def A285397_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-6*x+9*x^2)/(1-32*x+195*x^2-216*x^3) ).list()
A285397_list(40) # G. C. Greubel, Dec 09 2021

a(0) = 1, a(1) = 26, a(2) = 646, a(n) = 28*a(n-1) - 195*a(n-2) + 216*a(n-3).

G.f.: (1-6*x+9*x^2)/(1-32*x+195*x^2-216*x^3).

A285398 Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 0; a(n) is the number of cells after n iterations.

Original entry on oeis.org

1, 19, 452, 10948, 266300, 6484372, 157936172, 3847025764, 93707895260, 2282596837492, 55601016789068, 1354367059315396, 32990588541122684, 803607076375862356, 19574804963320797548, 476816346057854861860, 11614615234500986326556, 282916657894827156657460

Offset: 0

Peter Karpov, Apr 23 2017

Cell configuration converges to a fractal with dimension 2.906...

Colin Barker, Table of n, a(n) for n = 0..700
Peter Karpov, InvMem, Item 26
Peter Karpov, Illustration of initial terms (n = 1..4)
Index entries for linear recurrences with constant coefficients, signature (32,-195,216).

Cf. A007482, A026597, A285391, A285392, A285393, A285394, A285395, A285396, A285397, A285399, A285400.

I:=[19, 452, 10948]; [1] cat [n le 3 select I[n] else 32*Self(n-1) - 195*Self(n-2) + 216*Self(n-3) : n in [1..41]]; // G. C. Greubel, Dec 09 2021

{1}~Join~LinearRecurrence[{32, -195, 216}, {19, 452, 10948}, 17]

Vec((1 - x)*(1 - 3*x)*(1 - 9*x) / (1 - 32*x + 195*x^2 - 216*x^3) + O(x^20)) \\ Colin Barker, Apr 23 2017

def A285398_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-13*x+39*x^2-27*x^3)/(1-32*x+195*x^2-216*x^3) ).list()
A285398_list(40) # G. C. Greubel, Dec 09 2021

a(0) = 1, a(1) = 19, a(2) = 452, a(3) = 10948, a(n) = 28*a(n-1) - 195*a(n-2) + 216*a(n-3).

G.f.: (1-13*x+39*x^2-27*x^3)/(1-32*x+195*x^2-216*x^3).

A285399 Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 0 or 2; a(n) is the number of cells after n iterations.

Original entry on oeis.org

1, 13, 182, 2548, 35672, 499408, 6991712, 97883968, 1370375552, 19185257728, 268593608192, 3760310514688, 52644347205632, 737020860878848, 10318292052303872, 144456088732254208, 2022385242251558912, 28313393391521824768, 396387507481305546752

Offset: 0

Peter Karpov, Apr 23 2017

Cell configuration converges to a fractal with dimension 2.402...

Colin Barker, Table of n, a(n) for n = 0..850
Peter Karpov, InvMem, Item 26
Peter Karpov, Illustration of initial terms (n = 1..4)
Index entries for linear recurrences with constant coefficients, signature (14).

Cf. A007482, A026597, A285391, A285392, A285393, A285394, A285395, A285396, A285397, A285398, A285400.

[1] cat [13*14^(n-1): n in [1..40]]; // G. C. Greubel, Dec 09 2021

A285399:=n->13*14^(n-1): 1,seq(A285399(n), n=1..30); # Wesley Ivan Hurt, Apr 23 2017

{1}~Join~LinearRecurrence[{14}, {13}, 18]

Vec((1-x) / (1-14*x) + O(x^20)) \\ Colin Barker, Apr 23 2017

[1]+[13*14^(n-1) for n in (1..40)] # G. C. Greubel, Dec 09 2021

a(0) = 1, a(1) = 13, a(n) = 14*a(n-1).
G.f.: (1-x)/(1-14*x).
a(n) = 13 * 14^(n-1) for n>0. - Colin Barker, Apr 23 2017
E.g.f.: (1 + 13*exp(14*x))/14. - G. C. Greubel, Dec 09 2021

A285400 Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 0 or 3; a(n) is the number of cells after n iterations.

Original entry on oeis.org

1, 18, 378, 7938, 166698, 3500658, 73513818, 1543790178, 32419593738, 680811468498, 14297040838458, 300237857607618, 6304995009759978, 132404895204959538, 2780502799304150298, 58390558785387156258, 1226201734493130281418, 25750236424355735909778

Offset: 0

Peter Karpov, Apr 23 2017

Cell configuration converges to a fractal with dimension 2.771...

Colin Barker, Table of n, a(n) for n = 0..750
Peter Karpov, InvMem, Item 26
Peter Karpov, Illustration of initial terms (n = 1..4)
Index entries for linear recurrences with constant coefficients, signature (21).

Cf. A007482, A026597, A285391, A285392, A285393, A285394, A285395, A285396, A285397, A285398, A285399.

[1] cat [18*21^(n-1): n in [1..40]]; // G. C. Greubel, Dec 09 2021

{1}~Join~LinearRecurrence[{21}, {18}, 17]

Vec((1-3*x) / (1-21*x) + O(x^20)) \\ Colin Barker, Apr 23 2017

[1]+[18*21^(n-1) for n in (1..40)] # G. C. Greubel, Dec 09 2021

a(0) = 1, a(1) = 18, a(n) = 21*a(n-1).
G.f.: (1-3*x)/(1-21*x).
a(n) = 2 * 3^(n+1) * 7^(n-1) for n>0. - Colin Barker, Apr 23 2017
E.g.f.: (1 + 6*exp(21*x))/7. - G. C. Greubel, Dec 09 2021

A284395 Positions of 1 in A284394.

Original entry on oeis.org

2, 8, 17, 23, 32, 41, 47, 56, 62, 71, 80, 86, 95, 104, 110, 119, 125, 134, 143, 149, 158, 164, 173, 182, 188, 197, 206, 212, 221, 227, 236, 245, 251, 260, 269, 275, 284, 290, 299, 308, 314, 323, 329, 338, 347, 353, 362, 371, 377, 386, 392, 401, 410, 416, 425

Offset: 1

Clark Kimberling, May 02 2017

The sequences p = A032766, q = A285395, r = A284396 of positions of 0,1,2, respectively, partition the positive integers. Let t,u,v be the slopes of p, q, r, respectively. Then t = 3/2, u = (9+3*sqrt(5))/2, v = (3+3*sqrt(5))/2, and 1/t + 1/u + 1/v = 1.

			As a word, A284394 = 01002001002002001..., in which the positions of 1 are 2,8,17,...

Clark Kimberling, Table of n, a(n) for n = 1..10000

Cf. A003849, A032766, A284394, A284396.

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 13]  (* A003849 *)
w = StringJoin[Map[ToString, s]]; w1 = StringReplace[w, {"101" -> "2"}]
st = ToCharacterCode[w1] - 48 (* A284394 *)
Flatten[Position[st, 0]]  (* A032766 *)
Flatten[Position[st, 1]]  (* A284395 *)
Flatten[Position[st, 2]]  (* A284396 *)

cp's OEIS Frontend

A285391 Start with a single cell at coordinates (0, 0), then iteratively subdivide the grid into 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 2; a(n) is the number of cells after n iterations.

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A285392 Start with a single cell at coordinates (0, 0), then iteratively subdivide the grid into 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 0; a(n) is the number of cells after n iterations.

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A285393 Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 2 or 3; a(n) is the number of cells after n iterations.

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Programs

Magma

Mathematica

Sage

Formula

A285394 Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 0 or 1; a(n) is the number of cells after n iterations.

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Author

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Comments

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A285396 Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 2; a(n) is the number of cells after n iterations.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A285397 Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 3; a(n) is the number of cells after n iterations.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Sage

Formula

A285398 Start with a single cell at coordinates (0, 0, 0), then iteratively subdivide the grid into 3 X 3 X 3 cells and remove the cells whose sum of modulo 2 coordinates is 0; a(n) is the number of cells after n iterations.

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