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 1-4 of 4 results.

A296882 Numbers whose base-10 digits d(m), d(m-1), ..., d(0) have #(pits) = #(peaks); see Comments.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67
Offset: 1

Views

Author

Clark Kimberling, Jan 10 2018

Keywords

Comments

A pit is an index i such that d(i-1) > d(i) < d(i+1); a peak is an index i such that d(i-1) < d(i) > d(i+1). The sequences A296882-A296883 partition the natural numbers. See the guides at A296712. We have a(n) = A000027(n) for n=1..100 but not n=101.
.
Guide to related sequences:
Base #(pits) = #(peaks) #(pits) > #(peaks) #(pits) < #(peaks)
2 A296858 A296859 A296860
3 A296861 A296862 A296863
4 A296864 A296865 A296866
5 A296867 A296868 A296869
6 A296870 A296871 A296872
7 A296873 A296874 A296875
8 A296876 A296877 A296878
9 A296879 A296880 A296881
10 A296882 A296883 A296884
11 A296885 A296886 A296887
12 A296888 A296889 A296890
13 A296891 A296892 A296893
14 A296894 A296895 A296896
15 A296897 A296898 A296899
16 A296900 A296901 A296902
20 A296903 A296904 A296905
60 A296906 A296907 A296908

Examples

			The base-10 digits of 1212 are 1,2,1,2; here #(pits) = 1 and #(peaks) = 1, so 1212 is in the sequence.

Links

Crossrefs

Cf. A296882, A296712, A296883, A296884.

Programs

  • Mathematica
    z = 200; b = 10;
d[n_] := Differences[Sign[Differences[IntegerDigits[n, b]]]];
Select[Range [z], Count[d[#], -2] == Count[d[#], 2] &]  (* A296882 *)
Select[Range [z], Count[d[#], -2] < Count[d[#], 2] &]   (* A296883 *)
Select[Range [z], Count[d[#], -2] > Count[d[#], 2] &]   (* A296884 *)

Extensions

Overview table corrected by Georg Fischer, Aug 24 2021

A296901 Numbers whose base-16 digits d(m), d(m-1), ..., d(0) have #(pits) > #(peaks); see Comments.

Original entry on oeis.org

257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 769, 770, 771, 772, 773, 774, 775, 776
Offset: 1

Views

Author

Clark Kimberling, Jan 12 2018

Keywords

Comments

A pit is an index i such that d(i-1) > d(i) < d(i+1); a peak is an index i such that d(i-1) < d(i) > d(i+1). The sequences A296900-A296902 partition the natural numbers. See the guides at A296712 and A296882.

Examples

			The base-16 digits of 135698 are 2,1,2,1,2; here #(pits) = 2 and #(peaks) = 1, so 135698 is in the sequence.

Links

Crossrefs

Cf. A296882, A296712, A296900, A296902.

Programs

  • Mathematica
    z = 200; b = 16;
d[n_] := Differences[Sign[Differences[IntegerDigits[n, b]]]];
Select[Range [z], Count[d[#], -2] == Count[d[#], 2] &]  (* A296900 *)
Select[Range [z], Count[d[#], -2] < Count[d[#], 2] &]   (* A296901 *)
Select[Range [z], Count[d[#], -2] > Count[d[#], 2] &]   (* A296902 *)

A296902 Numbers whose base-16 digits d(m), d(m-1), ..., d(0) have #(pits) < #(peaks); see Comments.

Original entry on oeis.org

288, 289, 304, 305, 306, 320, 321, 322, 323, 336, 337, 338, 339, 340, 352, 353, 354, 355, 356, 357, 368, 369, 370, 371, 372, 373, 374, 384, 385, 386, 387, 388, 389, 390, 391, 400, 401, 402, 403, 404, 405, 406, 407, 408, 416, 417, 418, 419, 420, 421, 422, 423
Offset: 1

Views

Author

Clark Kimberling, Jan 12 2018

Keywords

Comments

A pit is an index i such that d(i-1) > d(i) < d(i+1); a peak is an index i such that d(i-1) < d(i) > d(i+1). The sequences A296900-A296902 partition the natural numbers. See the guides at A296712 and A296882.

Examples

			The base-16 digits of 74017 are 1,2,1,2,1; here #(pits) = 1 and #(peaks) = 2, so 74017 is in the sequence.

Links

Crossrefs

Cf. A296882, A296712, A296900, A296901.

Programs

  • Mathematica
    z = 200; b = 16;
d[n_] := Differences[Sign[Differences[IntegerDigits[n, b]]]];
Select[Range [z], Count[d[#], -2] == Count[d[#], 2] &]  (* A296900 *)
Select[Range [z], Count[d[#], -2] < Count[d[#], 2] &]   (* A296901 *)
Select[Range [z], Count[d[#], -2] > Count[d[#], 2] &]   (* A296902 *)

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 1-4 of 4 results.

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