A219158 -id:A219158 - cp's OEIS frontend

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

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.

A338861 a(n) is the largest area of a rectangle which can be dissected into n squares with integer sides s_i, i = 1 .. n, and gcd(s_1,...,s_n) = 1.

Original entry on oeis.org

1, 2, 6, 15, 42, 143, 399, 1190, 4209, 10920, 37245, 109886, 339745, 1037186, 3205734, 9784263, 29837784, 93313919, 289627536

Offset: 1

Rainer Rosenthal, Nov 12 2020

A219158 gives the minimum number of squares to tile an i x j rectangle. a(n) is found by checking all rectangles (i,j) for which A219158 has a dissection into n squares.

Due to the potential counterexamples to the minimal squaring conjecture (see MathOverflow link), terms after a(19) have to be considered only as lower bounds: a(20) >= 876696755, a(21) >= 2735106696. - Hugo Pfoertner, Nov 17 2020, Apr 02 2021

			a(6) = 11*13 = 143.
Dissection of the 11 X 13 rectangle into 6 squares:
.
          +-----------+-------------+
          |           |             |
          |           |             |
          |   6 X 6   |    7 X 7    |
          |           |             |
          |           |             |
          +---------+-+             |
          |         +-+-----+-------+
          |  5 X 5  |       |       |
          |         | 4 X 4 | 4 X 4 |
          |         |       |       |
          +---------+-------+-------+
.
a(19) = 16976*17061 = 289627536.
Dissection of the 16976 X 17061 rectangle into 19 squares:
.
       +----------------+-------------+
       |                |             |
       |                |             |
       |                |     7849    |
       |      9212      |             |
       |                |             |
       |                |             |
       |                |------+------|
       |________________|      |      |
       |             |   see   | 4109 |
       |             |Rosenthal|      |
       |             |  link +-+------+
       |     7764    |-------|        |
       |             |       |  5018  |
       |             | 4279  |        |
       |             |       |        |
       +-------------+-------+--------+
.

Stuart Anderson, Catalogues of Simple Perfect Squared Rectangles.
Stuart Anderson, Simple Imperfect Squared Rectangles, orders 9 to 24.
Bertram Felgenhauer, Filling rectangles with integer-sided squares.
MathOverflow, tiling a rectangle with the smallest number of squares, answer by Ed Pegg Jr, Jul 09 2017.
Rainer Rosenthal, Rectangle tiled by 19 squares with maximum area a(19)

Cf. A219158, A340726, A340920.

This sequence and A089047 are effectively analogs for dissecting (or tiling) rectangles and squares respectively. Analogs using equilateral triangular tiles are A014529 and A290821 respectively.

a(11)-a(17) from Hugo Pfoertner based on data from squaring.net website, Nov 17 2020
a(18) from Hugo Pfoertner, Feb 18 2021
a(19) from Hugo Pfoertner, Apr 02 2021

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.

A360030 a(n) is the minimum number of equal resistors needed in an electrical network so that n nodes can be selected in this network such that there are n*(n-1)/2 distinct resistances 0 < R < oo between the selected nodes.

Original entry on oeis.org

1, 3, 5, 8, 10, 11, 12, 14, 15, 16, 18, 19, 21

Offset: 2

Hugo Pfoertner and Rainer Rosenthal, Feb 12 2023

			a(2) = 1, [[1,2]]
.
  1           2
  O----R1R----O
  R_12 = 1
.
a(3) = 3, [[1,2]^2,[2,3]]
.
  1   .---R1R---.   2           3
  O --|         |-- O ---R3R--- O
      .---R2R---.
.
  R_12 = 1/2, R_13 = 3/2,
              R_23 = 1
.
a(4) = 5, node 5 hidden, [[1,2],[2,3]^2,[3,5],[4,5]]
.
  1           2   .---R2R---.   3          (5)          4
  O ---R1R--- O --|         |-- O ---R4R--- O ---R5R--- O
                  .---R3R---.
.
  R_12 = 1, R_13 = 3/2, R_14 = 7/2,
            R_23 = 1/2, R_24 = 5/2,
                        R_34 = 2
.
a(5) = 8, node 6 hidden,
  [[1, 2], [1, 3]^2, [2, 3], [2, 4], [3, 6], [4, 5], [4, 6]]
.
    1             2           4           5
    O-----R1R-----O----R5R----O----R8R----O
    |             |           |
    |            R4R         R7R
    .---R2R---.   |           |
    |         |---O----R6R----O
    .---R3R---.   3          (6)
.
   R_12 = 5/9, R_13 = 7/18, R_14 = 19/18, R_15 = 37/18,
               R_23 = 1/2,  R_24 = 13/18, R_25 = 31/18,
                            R_34 =  8/9,  R_35 = 17/9,
                                          R_45 =  1

IBM Research, Electric networks in graphs, Ponder This Challenge, March 2025, asked for the only network corresponding to a(10)=15 and 4 networks for a(12)=18.
Hugo Pfoertner, Illustrated examples for the terms a(6), a(7), a(8), 17 Feb 2023.
Hugo Pfoertner, Illustrated examples for the terms a(9), a(10), a(11), 3 Apr 2023.
Hugo Pfoertner, Illustration of a(12)=18, 8 Jan 2024, showing 3 planar and 5 non-planar networks, 4 of which were required to solve the bonus question of IBM's Ponder This Challenge.
Hugo Pfoertner and Klaus Nagel, Illustration of a(14)=21, 21 Aug 2025.

Cf. A219158, A342557, A342558, A348020.

a(14) from Klaus Nagel and Hugo Pfoertner, Aug 21 2025

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

A221839 Number of rectangles dissectible into a minimum of n squares.

Original entry on oeis.org

1, 1, 2, 4, 10, 22, 57, 157, 447

Offset: 1

Geoffrey H. Morley, Jan 26 2013

			For n = 4 the a(4) = 4 rectangles are 4X1, 4X3, 5X2 and 5X3.

Cf. A210517.

a(n) is the number of rectangles pXq for which GCD(p,q) = 1 and T(p,q) = n in A219158.

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Formula

A338861 a(n) is the largest area of a rectangle which can be dissected into n squares with integer sides s_i, i = 1 .. n, and gcd(s_1,...,s_n) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

A360030 a(n) is the minimum number of equal resistors needed in an electrical network so that n nodes can be selected in this network such that there are n*(n-1)/2 distinct resistances 0 < R < oo between the selected nodes.

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

A221839 Number of rectangles dissectible into a minimum of n squares.

Views

Author

Keywords

Examples

Crossrefs

Formula