A113881 -id:A113881 - cp's OEIS frontend

A226579 Smallest number of integer-sided squares needed to tile a 6 X n rectangle.

0, 6, 3, 2, 3, 5, 1, 5, 4, 3, 4, 6, 2, 6, 5, 4, 5, 7, 3, 7, 6, 5, 6, 8, 4, 8, 7, 6, 7, 9, 5, 9, 8, 7, 8, 10, 6, 10, 9, 8, 9, 11, 7, 11, 10, 9, 10, 12, 8, 12, 11, 10, 11, 13, 9, 13, 12, 11, 12, 14, 10, 14, 13, 12, 13, 15, 11, 15, 14, 13, 14, 16, 12, 16, 15, 14

Offset: 0

Alois P. Heinz, Jun 12 2013

			a(13) = 6:
._._._._._._._._._._._._._.
|           |       |     |
|           |       |     |
|           |       |_____|
|           |_______|     |
|           |   |   |     |
|___________|___|___|_____|

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

Row m=6 of A113881, A219158.

a:= n-> `if`(n=1, 6, iquo(n, 6, 'r') +[0, 4, 3, 2, 3, 5][r+1]):
seq(a(n), n=0..100);

Join[{0,6},LinearRecurrence[{0,1,1,0,-1},{3,2,3,5,1},80]] (* Harvey P. Dale, Jun 03 2014 *)

G.f.: x*(2*x^5-6*x^3-4*x^2+3*x+6)/(x^5-x^3-x^2+1).

a(n) = 1 + a(n-6) for n>7.

A226580 Smallest number of integer-sided squares needed to tile a 7 X n rectangle.

Original entry on oeis.org

0, 7, 5, 5, 5, 5, 5, 1, 7, 6, 6, 6, 6, 6, 2, 8, 7, 7, 7, 7, 7, 3, 9, 8, 8, 8, 8, 8, 4, 10, 9, 9, 9, 9, 9, 5, 11, 10, 10, 10, 10, 10, 6, 12, 11, 11, 11, 11, 11, 7, 13, 12, 12, 12, 12, 12, 8, 14, 13, 13, 13, 13, 13, 9, 15, 14, 14, 14, 14, 14, 10, 16, 15, 15, 15

Offset: 0

Alois P. Heinz, Jun 12 2013

			a(15) = 8:
._._._._._._._._._._._._._._._.
|             |       |       |
|             |       |       |
|             |       |       |
|             |_______|_______|
|             |     |     |   |
|             |     |     |___|
|_____________|_____|_____|_|_|

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

Row m=7 of A113881, A219158.

a:= n-> `if`(n=1, 7, iquo(n, 7, 'r') +[0, 6, 5$5][r+1]):
seq(a(n), n=0..100);

CoefficientList[Series[x*(x^8 - x^7 - 4*x^6 - 2*x + 7)/(x^8 - x^7 - x + 1), {x, 0, 100}], x] (* Wesley Ivan Hurt, Jan 15 2017 *)

G.f.: x*(x^8-x^7-4*x^6-2*x+7)/(x^8-x^7-x+1).

a(n) = 1 + a(n-7) for n>8.

A226581 Smallest number of integer-sided squares needed to tile an 8 X n rectangle.

Original entry on oeis.org

0, 8, 4, 5, 2, 5, 4, 7, 1, 7, 5, 6, 3, 6, 5, 8, 2, 8, 6, 7, 4, 7, 6, 9, 3, 9, 7, 8, 5, 8, 7, 10, 4, 10, 8, 9, 6, 9, 8, 11, 5, 11, 9, 10, 7, 10, 9, 12, 6, 12, 10, 11, 8, 11, 10, 13, 7, 13, 11, 12, 9, 12, 11, 14, 8, 14, 12, 13, 10, 13, 12, 15, 9, 15, 13, 14, 11

Offset: 0

Alois P. Heinz, Jun 12 2013

			a(17) = 8:
._._._._._._._._._._._._._._._._._.
|               |       |         |
|               |       |         |
|               |       |         |
|               |_______|         |
|               |       |_________|
|               |       |     |   |
|               |       |     |___|
|_______________|_______|_____|_|_|

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

Row m=8 of A113881, A219158.

a:= n-> `if`(n=1, 8, iquo(n, 8, 'r') +[0, 6, 4, 5, 2, 5, 4, 7][r+1]):
seq(a(n), n=0..100);

LinearRecurrence[{1,0,0,0,0,0,0,1,-1},{0,8,4,5,2,5,4,7,1,7,5},80] (* Harvey P. Dale, Sep 07 2016 *)

G.f.: x*(2*x^9-2*x^8-6*x^7+3*x^6-x^5+3*x^4-3*x^3+x^2-4*x+8) / (x^9-x^8-x+1).

a(n) = 1 + a(n-8) for n>9.

A226582 Smallest number of integer-sided squares needed to tile a 9 X n rectangle.

Original entry on oeis.org

0, 9, 6, 3, 6, 6, 3, 6, 7, 1, 6, 7, 4, 7, 7, 4, 7, 8, 2, 7, 8, 5, 8, 8, 5, 8, 9, 3, 8, 9, 6, 9, 9, 6, 9, 10, 4, 9, 10, 7, 10, 10, 7, 10, 11, 5, 10, 11, 8, 11, 11, 8, 11, 12, 6, 11, 12, 9, 12, 12, 9, 12, 13, 7, 12, 13, 10, 13, 13, 10, 13, 14, 8, 13, 14, 11, 14

Offset: 0

Alois P. Heinz, Jun 12 2013

			a(19) = 7:
._._._._._._._._._._._._._._._._._._._.
|                 |         |         |
|                 |         |         |
|                 |         |         |
|                 |         |         |
|                 |_________|_________|
|                 |       |       |   |
|                 |       |       |___|
|                 |       |       |   |
|_________________|_______|_______|___|

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

Row m=9 of A113881, A219158.

a:= n-> `if`(n=1, 9, iquo(n, 9, 'r')+[0, 5, 6, 3, 6, 6, 3, 6, 7][r+1]):
seq(a(n), n=0..100);

G.f.: x*(4*x^10-4*x^9-6*x^8+x^7+3*x^6-3*x^5+3*x^3-3*x^2-3*x+9) / (x^10-x^9-x+1).

a(n) = 1 + a(n-9) for n>10.

A226583 Smallest number of integer-sided squares needed to tile a 10 X n rectangle.

Original entry on oeis.org

0, 10, 5, 6, 4, 2, 4, 6, 5, 6, 1, 6, 5, 7, 5, 3, 5, 7, 6, 7, 2, 7, 6, 8, 6, 4, 6, 8, 7, 8, 3, 8, 7, 9, 7, 5, 7, 9, 8, 9, 4, 9, 8, 10, 8, 6, 8, 10, 9, 10, 5, 10, 9, 11, 9, 7, 9, 11, 10, 11, 6, 11, 10, 12, 10, 8, 10, 12, 11, 12, 7, 12, 11, 13, 11, 9, 11, 13, 12

Offset: 0

Alois P. Heinz, Jun 12 2013

			a(22) = 6:
._._._._._._._._._._._._._._._._._._._._._._.
|                   |           |           |
|                   |           |           |
|                   |           |           |
|                   |           |           |
|                   |           |           |
|                   |___________|___________|
|                   |       |       |       |
|                   |       |       |       |
|                   |       |       |       |
|___________________|_______|_______|_______|

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

Row m=10 of A113881, A219158.

a:= n-> `if`(n in [1, 2], 10/n, iquo(n, 10, 'r')+
    [0, 5, 4, 6, 4, 2, 4, 6, 5, 6][r+1]):
seq(a(n), n=0..100);

G.f.: x*(x^8+5*x^7-x^6-10*x^5-4*x^4-x^3-4*x^2+5*x+10) / (x^7-x^5-x^2+1).

a(n) = 1 + a(n-10) for n>12.

A348020 a(n) is the minimum number of unit resistors in a circuit with resistance R = A007305(n)/A007306(n).

Original entry on oeis.org

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

Offset: 1

Hugo Pfoertner, Oct 14 2021

nonn

For small values of n, the circuits are planar and correspond to the tiling of rectangles by squares. See A338573 for more information and examples.

The earliest nonplanar deviation occurs at a(3173) corresponding to R = 115/204 needing 11 instead of 12 resistors.

Hugo Pfoertner, Table of n, a(n) for n = 1..4096

First 31 terms coincide with A070941.
Cf. A007305, A007306, A113881, A219158, A338573, A347282.
A338579 can be used for a lookup of the position for a given rational R.

A340920 a(n) is the number of distinct resistances that can be produced from a planar circuit with exactly n unit resistors.

Original entry on oeis.org

1, 1, 2, 4, 9, 23, 57, 151, 427, 1263, 3807, 11549, 34843, 104459, 311317, 928719, 2776247, 8320757, 24967341, 74985337

Offset: 0

Hugo Pfoertner and Rainer Rosenthal, Feb 14 2021

			a(10) = 3807, whereas A337517(10) = 3823. The difference of 16 resistances results from the 15 terms of A338601/A338602 and the resistance 34/27 not representable by a planar network of 10 resistors, whereas it (but not 27/34) can be represented by a nonplanar network of 10 resistors.

Stuart Anderson, Catalogues of Simple Perfect Squared Rectangles (SPSR).
Stuart Anderson, Simple Imperfect Squared Rectangles, orders 9 to 24.

Cf. A002840, A048211, A174283, A180414, A337516, A337517, A338511, A338601, A338602, A340921.

Cf. A113881, A219158.

Replace the list of 3-connected graphs corresponding to A338511 by the list of planar graphs provided in A002840 in Andrew Howroyd's PARI program linked in A180414.

a(n) = A337517(n) for n <= 9, a(n) < A337517(n) for n >= 10.

a(19) from Hugo Pfoertner, Mar 15 2021

cp's OEIS Frontend

A226579 Smallest number of integer-sided squares needed to tile a 6 X n rectangle.

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A226580 Smallest number of integer-sided squares needed to tile a 7 X n rectangle.

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A226581 Smallest number of integer-sided squares needed to tile an 8 X n rectangle.

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A226582 Smallest number of integer-sided squares needed to tile a 9 X n rectangle.

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A226583 Smallest number of integer-sided squares needed to tile a 10 X n rectangle.

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Maple

Formula

A348020 a(n) is the minimum number of unit resistors in a circuit with resistance R = A007305(n)/A007306(n).

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A340920 a(n) is the number of distinct resistances that can be produced from a planar circuit with exactly n unit resistors.

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PARI

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