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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.

User: Thomas Young

Thomas Young's wiki page.

Thomas Young has authored 4 sequences.

A364607 Denominations of a 4-coin system that returns the fewest coins in change on average.

Original entry on oeis.org

1, 5, 18, 25
Offset: 1

Views

Author

Thomas Young, Aug 06 2023

Keywords

Comments

In 1995, Thomas Young, along with students Jeff Greenfield, Joe Culbert, and David Raabe, explored the idea of the best 4-coin system for making change for a dollar. They assumed that each value for change (from 1 cent to 99 cents) was equally likely and, using an original Turbo Pascal program, calculated the average number of coins returned for every possible set of 4 coin denominations. Their result was announced via the essay, "Change the Dime not the Dollar."
The average number of coins required to return 1..99 cents is A339333(99,4)/99 = 389/99. - Pontus von Brömssen, Aug 13 2023

Links

Crossrefs

Cf. A208953, A339333, A366013.

A363381 a(n) is the number of distinct n-cell patterns that tile an n X n square.

Original entry on oeis.org

1, 2, 1, 60, 1, 102, 1, 62714
Offset: 1

Views

Author

Thomas Young, May 30 2023

Keywords

Comments

Consider n unit squares contained within an n X n square. The n unit squares are an n-cell pattern of the n X n square if, when copied n-1 times, they can, with various rigid transformations, be combined to tessellate the n X n square.
Put another way:
Consider, for example, for n = 4, a transparency with an outline of a 4 X 4 square filled by 16 unit squares. Any 4 unit squares are painted the same color. Those four squares are a potential n-cell pattern of the 4 X 4 square. Three copies of the transparency are made with only the color of the 4 squares being different. If a person can stack the transparencies in such a way that they fill the 4 X 4 square, then the n-cell pattern is acceptable.
Put another way:
n unit squares from an n X n square are painted a color. Those n unit squares are an n-cell pattern. If n-1 copies of the pattern can be painted (each a different color) and together they fill the n X n square, then the n unit squares form an acceptable n-cell pattern.
Conjecture by Andrew Young: For an n X n square, where n is an odd prime, there is only one n-cell pattern.
Conjecture by Andrew Young and Thomas Young: An odd integer n>=3 is prime iff there exists only one n-cell pattern for an n X n square.
For any number n, there is always the 1 X n pattern that tiles the n X n square.
For composite numbers n = f1*f2, 1 < f1 <= f2 < n, there is always an additional f1 X f2 pattern. For example, a 14 X 14 square can be tiled using fourteen 1 X 14 rectangles or fourteen 2 X 7 rectangles; a 15 X 15 square can be tiled using fifteen 1 X 15 rectangles or fifteen 3 X 5 rectangles; a 9 X 9 square can be tiled using nine 1 X 9 rectangles or nine 3 X 3 squares (as in Sudoku!).
The second conjecture is a Corollary of the first: If n = p*q is not prime, then there is always a second tiling using rectangles, as explained above. Since the second conjecture implies the first, the two conjectures are actually equivalent. - M. F. Hasler, Jun 15 2025

Examples

			For n = 1, there is one 1-cell pattern because there is only one unit square to paint.
For n = 2, there are two 2-cell patterns:
   +---+---+     +---+---+         +---+
   | 1 | 2 |     | 1 | 2 |         | 1 |
   +---+---+     +---+---+   and   +---+---+
   | 3 | 4 |                           | 4 |
   +---+---+                           +---+
For n = 3, there is one 3-cell pattern:
   +---+---+---+
   | 1 | 2 | 3 |
   +---+---+---+
   | 4 | 5 | 6 |     It is   +---+---+---+
   +---+---+---+             | 1 | 2 | 3 |
   | 7 | 8 | 9 |             +---+---+---+
   +---+---+---+
For n = 4, there are sixty 4-cell patterns:
   +---+---+---+---+
   | 1 | 2 | 3 | 4 |   One is  +---+---+---+---+
   +---+---+---+---+           | 1 | 2 | 3 | 4 |
   | 5 | 6 | 7 | 8 |           +---+---+---+---+
   +---+---+---+---+
   | 9 |10 |11 |12 |     which is equivalent to:
   +---+---+---+---+                       +---+
   |13 |14 |15 |16 |                       | 1 |
   +---+---+---+---+                       +---+
                                           | 5 |
                                           +---+
and therefore these two are not            | 9 |
counted as distinct patterns.              +---+
                                           |13 |
                                           +---+
Another 4-cell pattern for a 4 X 4 square
   +---+---+---+---+
   | x | x | y | y |
   +---+---+---+---+   is
   | z | y | x | a |          +---+---+
   +---+---+---+---+          | x | x |
   | y | z | a | x |          +---+---+---+
   +---+---+---+---+                  | x |
   | a | a | z | z |                  +---+---+
   +---+---+---+---+                      | x |
                                          +---+
     +---+---+
     | x | x |
     +---+---+---+       is equivalent to
             | x |
             +---+---+
                 | x |
                 +---+
           +---+---+  +---+                          +---+
           | y | y |  | z |                          | a |
       +---+---+---+  +---+---+                  +---+---+
       | y |              | z |                  | a |
   +---+---+              +---+---+---+  +---+---+---+
   | y |                      | z | z |  | a | a |
   +---+                      +---+---+  +---+---+
because the shapes can be created through reflection, rotation, or translation.
Therefore, they are counted as one pattern.
For n = 5, there is one 5-cell pattern.

Links

Crossrefs

Cf. A291806, A227004, A291808, A360630, A291807, A328020, A220778.

Formula

a(n) >= 2 if n is composite.
For n > 1, a(n) = 1 iff n is an odd prime (conjectured: cf comments).

Extensions

a(7)-a(8) from Andrew Howroyd, Jun 04 2023
Minor edits by M. F. Hasler, Jun 15 2025

A178842 a(n) = binomial((n-1)^2, n).

Original entry on oeis.org

0, 0, 4, 126, 4368, 177100, 8347680, 450978066, 27540584512, 1878392407320, 141629804643600, 11703541346076580, 1052134368066259632, 102250849636865496528, 10683770265451303535424, 1194448077521704400002650, 142288257910903254700704000, 17993390003427864738863790640
Offset: 1

Views

Author

Thomas Young, Jun 17 2010

Keywords

Comments

Number of ways to place n objects in an (n-1) X (n-1) array (e.g., the number of ways to arrange stars in a flag's field pattern).

Links

Crossrefs

Cf. A014062 (binomial(n^2, n)).

Programs

  • GAP
    List([1..20], n -> Binomial((n-1)^2, n)); # G. C. Greubel, Jan 21 2019
  • Magma
    [Binomial((n-1)^2,n): n in [1..20]]; // G. C. Greubel, Jan 21 2019
  • Mathematica
    a[n_] := Binomial[(n - 1)^2, n]; Array[a, 18] (* Robert G. Wilson v, Jul 25 2010 *)
  • PARI
    vector(20, n, binomial((n-1)^2,n)) \\ G. C. Greubel, Jan 21 2019
  • Sage
    [binomial((n-1)^2,n) for n in (1..20)] # G. C. Greubel, Jan 21 2019

A179094 Disorder number of the n X n grid graph.

Original entry on oeis.org

0, 5, 23, 61, 119, 213, 335, 509, 719, 997, 1319, 1725, 2183, 2741, 3359, 4093, 4895
Offset: 1

Views

Author

Thomas Young, Jun 29 2010

Keywords

Comments

Old name: Fill an n X n array with various permutations of the integers 1, 2, 3, 4... n^2. Define the organization number of the n X n array to be the following: Start at 1, count the rectilinear steps to reach 2, then the rectilinear steps to reach 3, etc. Add them up. The array that has the maximum organization number would be the "most disorganized." This sequence is the sequence showing the most disorganized number for n X n arrays starting at 1 X 1.
Similar to sequence A047838.
My computer program worked as follows:
a) generate a permutation
b) place the permutation into the array
c) calculate the array position (row, column) of each integer
d) sort the integers into another array preserving row and column
e) travel the new array from 1..n^2 and summing the absolute value of the differences between the rows of consecutive integers and summing the absolute value of the differences of the columns of consecutive integers. The organization number is the sum of the two sums.
For instance, with the permutation 8, 3, 6, 5, 9, 1, 2, 7, 4 place the integers into a 3 X 3 array as such:
8 3 6
5 9 1
2 7 4
(Notice the next integer is a knight's move away. This is not the only sequence that will give an organization number of 23, but this is why I wonder if the sequence is the same as A098499.)
Then sort the integers preserving their row and column:
number, row, column
1, 2, 3
2, 3, 1
3, 1, 2
4, 3, 3
5, 2, 1
6, 1, 3
7, 3, 2
8, 1, 1
9, 2, 2
Traveling from 1 to 9, the differences in the row numbers are 1, 2, 2, 1, 1, 2, 2, 1 (a sum of 12) and the differences in the column numbers are 2, 1, 1, 2, 2, 1, 1, 1 (a sum of 11) therefore the organization number is 23.
This is basically a traveling salesman variant. - D. S. McNeil, Aug 26 2010

Links

Crossrefs

Cf. A047838, A098499.

Formula

A possible formula: a(n) = 0 for n=1, n^3-n-1 for odd n > 1, n^3-3 for even n? - D. S. McNeil, Aug 26 2010
Let b(n) correspond to McNeil's formula. Then b(n) <= a(n) <= b(n) + 1 (see link). - Sela Fried, Nov 28 2023
Empirical G.f.: x^2*(5+13*x+10*x^2-6*x^3+x^4+x^5)/((1-x)^4*(1+x)^2). - Colin Barker, Mar 29 2012

Extensions

a(3) corrected and a(4)-a(17) computed by D. S. McNeil, Aug 26 2010. D. S. McNeil also finds that a(19)=6839, a(21)=9239, a(23)=12143.
Edited by N. J. A. Sloane, Aug 26 2010
Typo in formula corrected by D. S. McNeil, Aug 26 2010
Equivalent simpler description from Eric W. Weisstein, Oct 08 2024

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