cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A130445 Integers in the range [1, 425] not expressible as a difference of the unique optimal Golomb ruler of order 24. See A130444.

Original entry on oeis.org

128, 137, 150, 159, 161, 165, 166, 169, 178, 183, 185, 187, 188, 194, 200, 202, 206, 207, 209, 218, 221, 222, 223, 225, 227, 230, 233, 236, 237, 238, 247, 250, 254, 257, 258, 259, 260, 264, 265, 266, 267, 268, 270, 272, 275, 276, 279, 280, 282, 284, 290
Offset: 1

Views

Author

Joseph Biberstine (jrbibers(AT)indiana.edu), May 26 2007

Keywords

Comments

This sequence has exactly 149 terms. The union of this with A130446 yields all integers in the range [1,425].

Examples

			128 is a term of this sequence between there do not exist integers 1 <= i < j <= 425 such that A130444(j) - A130444(i) = 128.

Links

Crossrefs

Cf. A130444: Main entry. A130446: Integers in [1, 425] expressible as a difference from A130444.

Programs

  • Mathematica
    A130444 = {0, 9, 33, 37, 38, 97, 122, 129, 140, 142, 152, 191, 205, 208, 252, 278, 286, 326, 332, 353, 368, 384, 403, 425}; Complement[Range[Last[A130444]], Sort[(A130444[[ #[[1]]]] - A130444[[ #[[2]]]]) & /@ Select[Tuples[Range[24], 2], #[[1]] > #[[2]] &]]]

A130444 Marking indices for the unique optimal Golomb ruler of order 24.

Original entry on oeis.org

0, 9, 33, 37, 38, 97, 122, 129, 140, 142, 152, 191, 205, 208, 252, 278, 286, 326, 332, 353, 368, 384, 403, 425
Offset: 1

Views

Author

Joseph Biberstine (jrbibers(AT)indiana.edu), May 26 2007

Keywords

Comments

By definition of optimal, there is no shorter Golomb ruler of order 24 (that is, a[24]-a[1] = 425 is minimal). Moreover, it is uniquely optimal. By definition of Golomb ruler, each difference from the sequence is unique. That is, for all 1 <= i < j <= 24 with a[j]-a[i] = d, we have a[y]-a[x] = d iff y=j and x=i. J. P. Robinson and A. J. Bernstein discovered this Golomb ruler in 1967. It was verified to be optimal on Nov 01 2004 by a 4-year computation on distributed.net that performed an exhaustive search through 555529785505835800 rulers. This ruler is not perfect because there are values not expressible as a difference of its terms. For these values, see A130445.

Examples

			a[5]-a[4] = 1. No other difference from the sequence gives 1.
a[10]-a[9] = 2. No other difference from the sequence gives 2.
a[5]-a[3] = 5. No other difference from the sequence gives 5.
No difference from the sequence gives, for example, 128. See A130445.

Links

Crossrefs

Cf. A130445: Integers in [1, 425] not expressible as a difference from this sequence. A130446: Integers in [1, 425] expressible as a difference from this sequence.

A373300 Sum of successive integers in a row of length p(n) where p counts integer partitions.

Original entry on oeis.org

1, 5, 15, 45, 105, 264, 555, 1221, 2445, 4935, 9324, 17941, 32522, 59400, 104808, 184569, 315711, 540540, 902335, 1504800, 2462724, 4014513, 6444425, 10316250, 16283707, 25610886, 39841865, 61720659, 94687230, 144731706, 219282679, 330996105, 495901413, 740046425
Offset: 1

Views

Author

Olivier Gérard, May 31 2024

Keywords

Comments

The length of each row is given by A000041.
As many sequences start like the positive integers, their row sums when disposed in this shape start with the same values.
Here is a sample list by A-number order of the sequences which are sufficiently close to A000027 to have the same row sums for at least 8 terms.
A069782, A088480, A090107, A090108, A090109, A115510, A115511, A130446, A131717, A132086, A153671, A167904, A296879, A296882, A296885, A296888, A296891, A296894, A296897, A296900, A296903, A296906, A303502, A317945, A335280, A361374.

Examples

			Let's put the list of integers in a triangle whose rows have length p(n), number of integer partitions of n.
.
    1 |  1
    5 |  2  3
   15 |  4  5  6
   45 |  7  8  9 10 11
  105 | 12 13 14 15 16 17 18
  264 | 19 20 21 22 23 24 25 26 27 28 29
  555 | 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44
.
The sequence gives the row sums of this triangle.

Crossrefs

Cf. A000027, seen as a triangle with shape A000041.
Cf. A373301, the same principle, but starting from integer zero instead of 1.
Cf. A006003, row sums of the integers but for the linear triangle.

Programs

  • Mathematica
    Module[{s = 0},
 Table[s +=
   PartitionsP[n - 1]; (s + PartitionsP[n])*(s + PartitionsP[n] - 1)/2 -
   s*(s - 1)/2, {n, 1, 30}]]
Showing 1-3 of 3 results.

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