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.

A237273 Triangle read by rows: T(n,k) = k+m, if k < m and k*m = n, or T(n,k) = k, if k^2 = n. Otherwise T(n,k) = 0. With n>=1 and 1<=k<=A000196(n).

Original entry on oeis.org

1, 3, 4, 5, 2, 6, 0, 7, 5, 8, 0, 9, 6, 10, 0, 3, 11, 7, 0, 12, 0, 0, 13, 8, 7, 14, 0, 0, 15, 9, 0, 16, 0, 8, 17, 10, 0, 4, 18, 0, 0, 0, 19, 11, 9, 0, 20, 0, 0, 0, 21, 12, 0, 9, 22, 0, 10, 0, 23, 13, 0, 0, 24, 0, 0, 0, 25, 14, 11, 10, 26, 0, 0, 0, 5
Offset: 1

Views

Author

Omar E. Pol, Feb 08 2014

Keywords

Comments

The first element of column k is in row k^2.
Column k lists k, k-1 zeros, and the positive integers but starting from 2*k+1 interleaved with k-1 zeros.
Row n has only one positive term iff n is a noncomposite number (A008578).
It appears that there are only eight rows that do not contain zeros. The indices of these rows are in A018253 (the divisors of 24).

Examples

			Triangle begins:
1;
3;
4;
5,   2;
6,   0;
7,   5;
8,   0;
9,   6;
10,  0,  3;
11,  7,  0;
12,  0,  0;
13,  8,  7;
14,  0,  0;
15,  9,  0;
16,  0,  8;
17, 10,  0,  4;
18,  0,  0,  0;
19, 11,  9,  0;
20,  0,  0,  0;
21, 12,  0,  9;
22,  0, 10,  0;
23, 13,  0,  0;
24,  0,  0,  0;
25, 14, 11, 10;
26,  0,  0,  0,   5;
27, 15,  0,  0,   0;
28,  0, 12,  0,   0;
29, 16,  0, 11,   0;
30,  0,  0,  0,   0;
31, 17, 13,  0,  11;
...
For n = 9 the divisors of n are 1, 3, 9, so row 9 is 10, 0, 3, because 1*9 = 9 and 3^2 = 9. The sum of row 9 is A000203(9) = 13.
For n = 12 the divisors of 12 are 1, 2, 3, 4, 6, 12, so row 12 is 13, 8, 7, because 1*12 = 12, 2*6 = 12 and 3*4 = 12. The sum of row 12 is A000203(12) = 28.

Links

Crossrefs

Row sums give A000203.
Row n has length A000196(n).
Column 1 is A065475.
Cf. A000290, A008578, A018253, A027750, A196020, A210959, A212119, A212120, A228812-A228814, A231347, A236104, A236631, A237519, A237593.

Programs

  • PARI
    T(n, k) = if (n % k, 0, if (k^2==n, k, k + n/k));
tabf(nn) = {for (n = 1, nn, v = vector(sqrtint(n), k, T(n, k)); print(v););} \\ Michel Marcus, Jun 19 2019

A236630 Irregular triangle T(n,k) of alternating sums of squares of entries in the rows in the triangle of A235791, read by rows.

Original entry on oeis.org

1, 4, 9, 8, 16, 15, 25, 21, 36, 32, 33, 49, 40, 41, 64, 55, 56, 81, 65, 69, 100, 84, 88, 87, 121, 96, 100, 99, 144, 119, 128, 127, 169, 133, 142, 141, 196, 160, 169, 165, 225, 176, 192, 188, 189, 256, 207, 223, 219, 220, 289, 225, 241, 237, 238
Offset: 1

Views

Author

Omar E. Pol, Jan 29 2014

Keywords

Comments

The original name was: Number of "ON" cells at n-th stage in a cellular automaton (or pseudo cellular automaton) related to sigma (see Comments for precise definition).

Examples

			Triangle begins:
    1;
    4;
    9,   8;
   16,  15;
   25,  21;
   36,  32,  33;
   49,  40,  41;
   64,  55,  56;
   81,  65,  69;
  100,  84,  88,  87;
  121,  96, 100,  99;
  144, 119, 128, 127;
  169, 133, 142, 141;
  196, 160, 169, 165;
  225, 176, 192, 188, 189;
  256, 207, 223, 219, 220;
  289, 225, 241, 237, 238;
  ...
From _Omar E. Pol_, Apr 20 2024: (Start)
Illustration of the 6th row as the area of a polygon (or the number of cells) in the fourth quadrant:
.     _ _ _ _ _ _       _ _ _ _ _ _       _ _ _ _ _ _
.    |           |     |           |     |           |
.    |           |     |           |     |           |
.    |           |     |           |     |           |
.    |           |     |        _ _|     |          _|
.    |           |     |       |         |        _|
.    |_ _ _ _ _ _|     |_ _ _ _|         |_ _ _ _|
.
.          36           36 - 4 = 32     36 - 4 + 1 = 33
.
(End)

Links

Crossrefs

Row n has length A003056(n).
The first element of column k is in row A000217(k).
Column 1 gives the positive terms of A000290.
Right border gives A024916.
Row n is the alternating sum of entries in row n of A236104.
Cf. A003056, A000203, A004125, A026914, A071561, A196020, A235791, A236631, A237048, A237591, A237593, A237270, A249223, A264116, A280851.

Programs

  • Mathematica
    Map[Accumulate, Table[(-2 Boole[EvenQ[k]] + 1)*Ceiling[(n + 1)/k - (k + 1)/2]^2, {n, 20}, {k, Floor[(Sqrt[8*n + 1] - 1)/2]}]] // Flatten (* Michael De Vlieger, Apr 30 2024, after Hartmut F. W. Hoft at A235791 *)

Formula

From Hartmut F. W. Hoft, Apr 30 2024: (Start)
T(n, k) = Sum_{j = 1 .. k} (-1)^(j + 1) * S(n, j)^2, n >= 0, 1 <= k <= A003056(n), where S(n, j) is the j-th entry in the n-th row of the triangle of A235791.
T(n, k) = Sum_{j = 1 .. k} (-1)^(j+1) * S(n, j), n >= 0, 1 <= k <= A003056(n), where S(n, j) is the j-th entry in the n-th row of the triangle of A236104. (End)

Extensions

New name from Hartmut F. W. Hoft, Apr 27 2024
0 removed, offset changed and minor edits from Omar E. Pol, Apr 28 2024

A236109 Triangle read by rows: another version of A048158, only here the representation of A004125 is symmetric, as in the representation of A024916 and A000203.

Original entry on oeis.org

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

Views

Author

Omar E. Pol, Jan 26 2014

Keywords

Comments

Row sums give A004125.
For more information see A236104, A237591, A237593, A237270.

Examples

			Triangle begins:
0;
0, 0;
0, 0, 1;
0, 0, 0, 1;
0, 0, 0, 2, 2;
0, 0, 0, 0, 1, 2;
0, 0, 0, 0, 2, 3, 3;
0, 0, 0, 0, 0, 2, 3, 3;
0, 0, 0, 0, 0, 2, 2, 4, 4;
0, 0, 0, 0, 0, 0, 2, 3, 4, 4;
0, 0, 0, 0, 0, 0, 3, 4, 5, 5, 5;
0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 5, 5;
...
For the symmetric representation of A000203, A024916, A004125 in the fourth quadrant using a diagram which arises from the sequence A236104 see below:
--------------------------------------------------
n     A000203  A024916            Diagram
--------------------------------------------------
.                         _ _ _ _ _ _ _ _ _ _ _ _
1        1        1      |_| | | | | | | | | | | |
2        3        4      |_ _|_| | | | | | | | | |
3        4        8      |_ _|  _|_| | | | | | | |
4        7       15      |_ _ _|    _|_| | | | | |
5        6       21      |_ _ _|  _|  _ _|_| | | |
6       12       33      |_ _ _ _|  _| |  _ _|_| |
7        8       41      |_ _ _ _| |_ _|_|    _ _|
8       15       56      |_ _ _ _ _|  _|     |* *
9       13       69      |_ _ _ _ _| |      _|* *
10      18       87      |_ _ _ _ _ _|  _ _|* * *
11      12       99      |_ _ _ _ _ _| |* * * * *
12      28      127      |_ _ _ _ _ _ _|* * * * *
.
The 12th row is ........ 0,0,0,0,0,0,0,2,2,3,5,5
.
The total number of cells in the first n set of symmetric regions of the diagram equals A024916(n). It appears that the total number of cells in the n-th set of symmetric regions of the diagram equals sigma(n) = A000203(n). Example: for n = 12 the 12th row of triangle is 144, 25, 9, 1, hence the alternating sums is 144 - 25 + 9 - 1 = 127. On the other hand we have that A000290(12) - A004125(12) = 144 - 17 = A024916(12) = 127, equaling the total number of cells in the diagram after 12 stages. The number of cells in the 12th set of symmetric regions of the diagram is sigma(12) = A000203(12) = 28. Note that in this case there is only one region. The number of "*"'s is A004125(12) = 17.

Crossrefs

Cf. A000203, A004125, A024916, A048158, A196020, A235799, A236104, A236630, A236631, A237591, A237593, A237270.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.