Peter Karpov - cp's OEIS frontend

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.

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

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

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).

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).

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).

A285395 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 1; a(n) is the number of cells after n iterations.

Original entry on oeis.org

1, 15, 249, 4371, 78693, 1431735, 26159649, 478778379, 8768545197, 160633207887, 2942987702217, 53921188767939, 987955842925365, 18101659807680423, 331665550405417905, 6076909441961837115, 111343619823563404797, 2040083652208572498111, 37379255055440801236953

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, A285396, A285397, A285398, A285399, A285400.

I:=[1,15,249]; [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, 15, 249}, 20]

def A285395_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-13*x+24*x^2)/(1-28*x+195*x^2-324*x^3) ).list()
A285395_list(40) # G. C. Greubel, Dec 10 2021

a(0) = 1, a(1) = 15, a(2) = 249, a(n) = 28*a(n-1) - 195*a(n-2) + 324*a(n-3).

G.f.: (1-13*x+24*x^2)/(1-28*x+195*x^2-324*x^3).

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

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

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

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.

Original entry on oeis.org

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

cp's OEIS Frontend

User: Peter Karpov

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.

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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.

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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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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.

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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.

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A285395 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 1; a(n) is the number of cells after n iterations.

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Magma

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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.