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 1-5 of 5 results.

A167384 Irregular table with the left half of the array described in A167381.

Original entry on oeis.org

1, 3, 5, 6, 9, 10, 13, 14, 17, 18, 21, 22, 23, 27, 28, 29, 33, 34, 35, 39, 40, 41, 45, 46, 47, 51, 52, 53, 57, 58, 59, 60, 65, 66, 67, 68, 73, 74, 75, 76, 81, 82, 83, 84, 89, 90, 91, 92, 97, 98, 99, 100
Offset: 0

Views

Author

Paul Curtz, Nov 02 2009

Keywords

Examples

			1;
3;
5,6;
9,10;
13,14;
17,18;
21,22,23;
27,28,29;
33,34,35;
39,40,41;

Crossrefs

Cf. A001670 (number of terms per row), A167381 (last term of row n), A167413.

Formula

T(n,k) = T(n,k-1)+1, k>=1.

A284384 Alternate A167381(n), A167381(n) + 1.

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 6, 7, 10, 11, 14, 15, 18, 19, 23, 24, 29, 30, 35, 36, 41, 42, 47, 48, 53, 54, 60, 61, 68, 69, 76, 77, 84, 85, 92, 93, 100, 101, 108, 109, 116, 117, 125, 126, 135, 136, 145, 146, 155, 156, 165, 166, 175, 176, 185, 186, 195, 196, 205, 206, 215, 216
Offset: 0

Views

Author

Paul Curtz, Mar 26 2017

Keywords

Comments

The first differences (a(n+1) - a(n)) are b(n) = 1, 0, 1, 1, 1, 2, 1, 3, 1, 3, 1, 3, 1, 4, 1, 5, 1, 5, 1, 5, 1, 5, 1, 5, 1, 6, 1, 7, 1, 7, ... .
b(2n+1) = 0, 1, 2, 3, 3, 3, 4, 5, 5, 5, 5, 5, 6, 7, 7, 7, 7, 7, 7, 7, 8, ... =
0, 1,
2, 3, 3, 3
4, 5, 5, 5, 5, 5,
6, 7, 7, 7, 7, 7, 7, 7,
...
= 2*n followed by 2*n+1 times 2*n+1 (by rows) = A282692(n)?
= A284359(n) - 1.

Crossrefs

Cf. A005408, A167381, A282692, A284359.

A167991 Blocks of size 2n, each with 2n-1 replicas of 2n followed by 2n+1; n=1, 2, 3, ...

Original entry on oeis.org

2, 3, 4, 4, 4, 5, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 18, 18, 18, 18
Offset: 1

Views

Author

Paul Curtz, Nov 16 2009

Keywords

Comments

First differences of A167381.
The sum of the terms in block n is 4*n^2+1 = A053755(n).

Examples

			(2, 3), (4, 4, 4, 5), (6, 6, 6, 6, 6, 7), (8, 8, 8, 8, 8, 8, 8, 9), ...

Programs

  • Mathematica
    r[1] = Range[4];
r[n_] := r[n] = Range[r[n-1][[-1]]+1, r[n-1][[-1]] + (2n)^2];
s[n_] := Partition[r[n], Sqrt[Length[r[n]]]][[All, n]];
A167991 = Table[s[n], {n, 1, 9}] // Flatten // Differences (* Jean-François Alcover, Mar 27 2017 *)

A284359 Double triangle (2*n+2 terms by row). Every row is 2*n + 1 followed by 2*n + 1 times 2*n + 2.

Original entry on oeis.org

1, 2, 3, 4, 4, 4, 5, 6, 6, 6, 6, 6, 7, 8, 8, 8, 8, 8, 8, 8, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 13, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 14, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17
Offset: 0

Views

Author

Paul Curtz, Mar 25 2017

Keywords

Comments

In essence the same as A167991. - R. J. Mathar, Mar 27 2017

Examples

			1,  2,
3,  4,  4,  4,
5,  6,  6,  6,  6,  6,
7,  8,  8,  8,  8,  8,  8,  8,
9, 10, 10, 10, 10, 10, 10, 10, 10, 10,
... .
The row sum is A000466(n+1).

Crossrefs

Cf. A000466, A005408, A103517 (main diagonal), A167381.

Programs

  • Mathematica
    Table[2 n + 2 - Boole[k == 1], {n, 0, 8}, {k, 2 n + 2}] // Flatten (* Michael De Vlieger, Mar 25 2017 *)
  • PARI
    for(n=0, 10, for(k=1, 2*n + 2, print1(2*n + 2 - (k==1), ", ");); print();) \\ Indranil Ghosh, Mar 26 2017, translated from Mathematica code
  • Python
    for n in range(0, 11):
    print([2*n + 2 -(k==1) for k in range(1, 2*n + 3)])
# Indranil Ghosh, Mar 26 2017

Formula

a(n) = A167381(n+1) - A167381(n).

A167413 Irregular array with the first differences of row A167384(n,.) in row n.

Original entry on oeis.org

2, 2, 1, 3, 1, 3, 1, 3, 1, 3, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 4, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 5, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6
Offset: 0

Views

Author

Paul Curtz, Nov 03 2009

Keywords

Comments

The left column has 2 two's, 4 three's, 6 four's,... ,2*k-2 k's.
Entries in the remaining columns are all 1.
If one looks at the flattened sequence (2), (2), (1,3), (1,3), (1,3),.. in blocks of 1, 1, 2, 2, 2,.. , A000194(k) terms, the sum of entries in the subsequence in block number k is A001670(k).

Examples

			2;
2, 1;
3, 1;
3, 1;
3, 1;
3, 1, 1;
4, 1, 1;
4, 1, 1;
4, 1, 1;
4, 1, 1;
4, 1, 1;
4, 1, 1, 1;
5, 1, 1, 1;
5, 1, 1, 1;
5, 1, 1, 1;
5, 1, 1, 1;
5,...

Crossrefs

Cf. A167381.
Showing 1-5 of 5 results.

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