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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A279317 Minimal number of squares in a dissection of an (n) X (n+1) oblong into squares.

Original entry on oeis.org

2, 3, 4, 5, 5, 5, 7, 7, 6, 6, 7, 7, 7, 7, 7, 8, 8, 7, 9, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 8, 9, 9, 9, 8, 9, 9, 9, 9, 9, 9, 9, 9, 9, 10, 9, 9, 10, 9, 10, 9, 10, 10, 10, 10, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 10, 10, 10, 10, 10, 10, 10, 11, 11, 10, 10, 10, 10, 10, 10, 11, 10, 11, 10, 11, 10, 11, 11, 11, 11, 10, 11, 11, 11, 11, 11, 11, 11, 11, 10, 11, 11, 11, 11, 11, 11, 11, 11, 12
Offset: 1

Views

Author

Ed Pegg Jr, Dec 09 2016

Keywords

Comments

This is very close to b(n) = round(n^(1/3)) + 6. b(18)-a(18) = 2. b(387)-a(387) = 0. All b(n)-a(n) terms in between these points are -1, 0, 1.
Bouwkamp codes of dissections that are believed to be optimal follow.
10 105 104 60 45 19 26 44 16 12 7 33 28
11 177 176 99 78 21 57 77 43 16 41 34 9 25
12 308 307 165 143 22 67 54 142 45 13 41 97 28 69
13 552 551 312 240 44 60 136 28 16 76 239 101 37 175 138
14 970 969 546 424 172 252 423 73 50 23 119 80 96 39 293 254
15 1699 1698 951 748 307 441 747 127 77 50 27 200 134 177 66 509 443
16 2926 2925 1633 1293 213 299 781 127 86 41 344 1292 509 206 138 68 851 783
17 5211 5210 2846 2365 571 518 1276 2364 392 90 53 465 302 412 694 584 293 1569 1278
18 8731 8730 4741 3990 751 1195 2044 3989 1059 444 790 849 884 175 709 256 197 2696 453 2046
19 15131 15130 8169 6962 2415 4547 6961 1208 1943 1680 263 965 452 1504 702 1378 3621 802 865 2306 2243
20 25679 25678 13719 11960 1456 1866 2626 6012 303 743 410 11959 1623 440 1516 760 1183 3386 4322 1692 7706 6014
21 49583 49582 27252 22331 4763 5036 12532 158 4332 273 22330 5080 5309 906 2176 1250 4716 1270 4372 2187 3446 14719 12534

Examples

			Oblong 18 X 19 uses 7 squares of size 3, 5, 5, 7, 7, 8, 11.
Oblong 34 X 35 uses 8 squares of size 4, 7, 9, 9, 11, 15, 16, 19.
Oblong 55 X 56 uses 9 squares of size 5, 9, 12, 12, 14, 19, 23, 24, 32.
Oblong 104 X 105 uses 10 squares of size 7, 12, 16, 19, 26, 28, 33, 44, 45, 60.
From _Peter Kagey_, Dec 13 2016: (Start)
An example of the a(10) = 6 squares that can dissect a 10 X 11 oblong:
  +-------+-----------+
  |       |           |
  |   4   |           |
  |       |     6     |
  +---+---+           |
  | 2 | 2 |           |
  +---+---+-+---------+
  |         |         |
  |    5    |    5    |
  |         |         |
  |         |         |
  +---------+---------+
(End)

Links

Crossrefs

Cf. A005670, A339548.

Extensions

Corrected term 351 and extended to n=387 by Ed Pegg Jr, Oct 31 2018

A339808 1 + 1/a(n) is the smallest resistance value of this form that can be obtained from a resistor network of not more than n one-ohm resistors.

Original entry on oeis.org

1, 2, 3, 6, 10, 18, 34, 55, 104, 176, 320, 592, 1071, 1855, 3311, 5943, 10231, 19087
Offset: 2

Views

Author

Hugo Pfoertner, Dec 18 2020

Keywords

Examples

			a(2) = 1: 2 resistors in series produce a resistance of 2 = 1 + 1/a(1) ohm.
a(3) = 2: 3 resistors can produce {1/3, 2/3, 3/2, 3} ohms. The smallest resistance > 1 is 3/2 = 1 + 1/a(2) ohms.
a(4) = 3: 4 resistors can produce the A337517(4) = 9 distinct resistances {1/4, 2/5, 3/5, 3/4, 1, 4/3, 5/3, 5/2, 4} of which 4/3 = 1 + 1/a(4) is the smallest resistance > 1 ohm.
a(n) first differs from A339548(n) - 1 for n = 13. The resistance values of the A337517(13) = 110953 distinct resistances that can be obtained from a network of exactly 13 one-ohm resistors closest to 1 ohm are { ..., 551/552, 576/577, 596/597, 609/610, 1, 593/592, 580/579, 552/551, ...}. The largest resistance < 1 of a network of 13 one-ohm resistors is 609/610 = 1 - 1/A339548(13) ohms, whereas the smallest resistance > 1 is 593/a(13) = 593/592 ohms.
The resistor networks from which the target resistance R = 1 + 1/a(n) can be obtained correspond to simple or multigraphs whose edges are one-ohm resistors. Parallel resistors on one edge are indicated by an exponent > 1 after the affected vertex pair. The resistance R occurs between vertex number 1 and the vertex with maximum number in the graph. In some cases there are other possible representations in addition to the representation given.
.
resistors     vertices
   |      R       |   edges
   2     2/1      2  [1,2],[2,3]
   3     3/2      3  [1,2]^2,[2,3]
   4     4/3      4  [1,2]^3,[2,3]
   5     7/6      4  [1,2]^2,[2,3],[2,4],[3,4]
   6    11/10     5  [1,2]^2,[2,3]^2,[2,4],[3,4]
   7    19/18     5  [1,2]^2,[1,3],[2,3],[2,4],[3,5],[4,5]
   8    35/34     6  [1,2]^2,[1,3],[2,3],[2,4],[3,4],[3,5],[4,5]
   9    56/55     6  [1,2],[1,3],[1,4],[2,4],[3,4],[3,5],[4,5],[4,6],[5,6]
  10   105/104    7  [1,3],[1,4],[1,5],[2,4],[2,5],[2,6],[3,4],[3,5],[4,5],
                     [4,6]
  11   177/176    7  [1,2],[1,4],[1,6],[2,6],[2,7],[3,5],[3,6],[3,7],[4,5],
                     [4,6],[5,6]
  12   321/320    7  [1,2],[1,4],[1,5],[2,5],[2,6],[3,5],[3,6],[3,7],[4,5],
                     [4,6],[4,7],[5,6]
  13   593/592    8  [1,4],[1,5],[1,7],[2,4],[2,5],[2,6],[2,7],[3,5],[3,6],
                     [3,7],[3,8],[4,6],[5,8]
  14  1072/1071   9  [1,6],[1,8],[2,7],[2,8],[2,9],[3,5],[3,7],[3,9],[4,6],
                     [4,7],[4,8],[5,6],[5,8],[6,9]
  15  1856/1855   9  [1,5],[1,7],[2,5],[2,6],[2,7],[2,8],[3,6],[3,7],[3,9],
                     [4,6],[4,8],[4,9],[5,8],[5,9],[7,8]
  16  3312/3311  10  [1,7],[1,9],[2,6],[2,7],[2,8],[3,7],[3,8],[3,9],[4,5],
                     [4,6],[4,10],[5,8],[5,9],[6,9],[7,10],[8,10]
  17  5944/5943  10  [1,2],[1,3],[2,4],[2,5],[2,7],[3,4],[3,6],[3,10],[4,6],
                     [4,8],[5,6],[5,8],[6,9],[7,8],[7,9],[8,10],[9,10]
  18 10232/10231 11  [1,2],[1,3],[2,4],[2,6],[2,7],[3,5],[3,8],[3,11],[4,5],
                     [4,9],[5,7],[6,8],[6,9],[7,9],[7,10],[8,10],[9,11],[10,11]
  19 19088/19087 11  [1,2],[1,3],[2,4],[2,5],[2,7],[3,5],[3,6],[3,11],[4,6],
                     [4,9],[5,8],[5,10],[6,7],[6,8],[7,9],[7,10],[8,9],[9,11],[10,11]

Links

Crossrefs

Cf. A180414, A337517, A339548.

Extensions

a(18) from Hugo Pfoertner, Apr 09 2021
a(19) from Fedor Karpelevitch, Aug 20 2025

A340708 Maximum denominator of resistances obtained by an electrical network with n unit resistors.

Original entry on oeis.org

1, 2, 3, 5, 8, 13, 24, 40, 69, 130, 231, 408, 689, 1272, 2153, 3960, 6993, 12560
Offset: 1

Views

Author

Rainer Rosenthal, Jan 16 2021

Keywords

Comments

a(n) is taken from the set of resistance values counted by A337517(n). These sets can be computed by the PARI program of Andrew Howroyd in A180414.
Also the maximum numerator of these electrical networks for small n.
Maximum numerator and maximum denominator coincide for planar networks: for every resistance R in a planar network with n resistors there is always another planar network with n resistors and resistance 1/R. For nonplanar networks this is not necessarily so, as can be seen in A338573.
The asymmetry is illustrated by the example a(15) = 2153.
The author conjectures that this asymmetry will increase with n, and eventually the maxima will differ.
Conjecture: a(19) = 22233, a(20) = 39918. It would be very desirable to know at which value of n > 18 the maximum values of numerators and denominators differ for the first time. - Hugo Pfoertner, Apr 19 2021

Examples

			Denominators for numerator a(15) = 2153 in electrical networks with 15 resistors:
  1025,1049,1051,1058,1089,1104,1145,1184,1185,1193,1208,
  1212,1219,1248,1254,1337,1382,1403,1526,1527,1529,1530,
  1545,1547,1555,1579,1586,1632,1642,1647,1687,1699,1719.
Numerators for denominator a(15) = 2153 in electrical networks with 15 resistors:
   899, 905, 934, 941, 945, 960, 968, 969,1008,1049,1064,
  1095,1102,1104,1128,1137,1143,1147,1164,1182,1207,1296,
  1359,1367,1387,1400,1447,1543.

Links

Crossrefs

Cf. A180414, A337517, A338601, A339548, A339808, A340726.

Extensions

a(18) from Hugo Pfoertner, Apr 11 2021
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