A332266
Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists successive blocks of k consecutive integers that differ by 2, where the m-th block starts with m, m >= 1, and the first element of column k is in row k^2.
Original entry on oeis.org
1, 2, 3, 4, 1, 5, 3, 6, 2, 7, 4, 8, 3, 9, 5, 1, 10, 4, 3, 11, 6, 5, 12, 5, 2, 13, 7, 4, 14, 6, 6, 15, 8, 3, 16, 7, 5, 1, 17, 9, 7, 3, 18, 8, 4, 5, 19, 10, 6, 7, 20, 9, 8, 2, 21, 11, 5, 4, 22, 10, 7, 6, 23, 12, 9, 8, 24, 11, 6, 3, 25, 13, 8, 5, 1, 26, 12, 10, 7, 3, 27, 14, 7, 9, 5
Offset: 1
Triangle begins:
1;
2;
3;
4, 1;
5, 3;
6, 2;
7, 4;
8, 3;
9, 5, 1;
10, 4, 3;
11, 6, 5;
12, 5, 2;
13, 7, 4;
14, 6, 6;
15, 8, 3;
16, 7, 5, 1;
17, 9, 7, 3;
18, 8, 4, 5;
19, 10, 6, 7;
20, 9, 8, 2;
21, 11, 5, 4;
22, 10, 7, 6;
23, 12, 9, 8;
24, 11, 6, 3;
25, 13, 8, 5, 1;
...
Figures A..G show the location (in the columns of the table) of the partitions of n = 1..7 (respectively) into consecutive parts that differ by 2:
. ---------------------------------------------------------
Fig: A B C D E F G
. ---------------------------------------------------------
. n: 1 2 3 4 5 6 7
Row ---------------------------------------------------------
1 | [1];| 1; | 1; | 1; | 1; | 1; | 1; |
2 | | [2];| 2; | 2; | 2; | 2; | 2; |
3 | | | [3];| 3; | 3; | 3; | 3; |
4 | | | | [4],[1];| 4, 1;| 4, 1; | 4, 1;|
5 | | | | 5, [3];| [5], 3;| 5, 3; | 5, 3;|
6 | | | | | | [6],[2];| 6, 2;|
7 | | | | | | 7, [4];| [7], 4;|
. ---------------------------------------------------------
Figure F: for n = 6 the partitions of 6 into consecutive parts that differ by 2 (but with the parts in increasing order) are [6] and [2, 4]. These partitions have one part and two parts respectively. On the other hand we can find the mentioned partitions in the columns 1 and 2 of this table, starting at the row 6.
.
Figures H..L show the location (in the columns of the table) of the partitions of 8..12 (respectively) into consecutive parts that differ by 2:
. -----------------------------------------------------------------------
Fig: H I J K L
. -----------------------------------------------------------------------
. n: 8 9 10 11 12
Row -----------------------------------------------------------------------
1 | 1; | 1; | 1; | 1; | 1; |
1 | 2; | 2; | 2; | 2; | 2; |
3 | 3; | 3; | 3; | 3; | 3; |
4 | 4, 1; | 4, 1; | 4, 1; | 4, 1; | 4, 1; |
5 | 5, 3; | 5, 3; | 5, 3; | 5, 3; | 5, 3; |
6 | 6, 2; | 6, 2; | 6, 2; | 6, 2; | 6, 2; |
7 | 7, 4; | 7, 4; | 7, 4; | 7, 4; | 7, 4; |
8 | [8],[3]; | 8, 3; | 8, 3; | 8, 3; | 8, 3; |
9 | 9, [5], 1;| [9], 5, [1];| 9, 5, 1;| 9, 5, 1;| 9, 5, 1; |
10 | | 10, 4, [3];| [10],[4], 3;| 10, 4, 3;| 10, 4; 3; |
11 | | 11, 6, [5];| 11, [6], 5;| [11], 6, 5,| 11, 6; 5; |
12 | | | | | [12],[5],[2];|
13 | | | | | 13, [7],[4];|
14 | | | | | 14, 6, [6];|
. -----------------------------------------------------------------------
Figure I: for n = 9 the partitions of 9 into consecutive parts that differ by 2(but with the parts in increasing order) are [9] and [1, 3, 5]. These partitions have one part and three parts respectively. On the other hand, we can find the mentioned partitions in the columns 1 and 3 of this table, starting at the row 9.
.
Illustration of initial terms arranged into a triangular structure:
. _
. _|1|
. _|2 |
. _|3 _|
. _|4 |1|
. _|5 _|3|
. _|6 |2 |
. _|7 _|4 |
. _|8 |3 _|
. _|9 _|5 |1|
. _|10 |4 |3|
. _|11 _|6 _|5|
. _|12 |5 |2 |
. _|13 _|7 |4 |
. _|14 |6 _|6 |
. _|15 _|8 |3 _|
. _|16 |7 |5 |1|
. _|17 _|9 _|7 |3|
. _|18 |8 |4 |5|
. _|19 _|10 |6 _|7|
. _|20 |9 _|8 |2 |
. _|21 _|11 |5 |4 |
. _|22 |10 |7 |6 |
. _|23 _|12 _|9 _|8 |
. _|24 |11 |6 |3 _|
. |25 |13 |8 |5 |1|
...
The number of horizontal line segments in the n-th row of the diagram equals A038548(n), the number of partitions of n into consecutive parts that differ by 2.
Tables of the same family where the consecutive parts differ by d are
A010766 (d=0),
A286001 (d=1), this sequence (d=2),
A334945 (d=3),
A334618(d=4).
A334618
Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists successive blocks of k consecutive integers that differ by 4, where the m-th block starts with m, m >= 1, and the first element of column k is in the row that is the k-th hexagonal number (A000384).
Original entry on oeis.org
1, 2, 3, 4, 5, 6, 1, 7, 5, 8, 2, 9, 6, 10, 3, 11, 7, 12, 4, 13, 8, 14, 5, 15, 9, 1, 16, 6, 5, 17, 10, 9, 18, 7, 2, 19, 11, 6, 20, 8, 10, 21, 12, 3, 22, 9, 7, 23, 13, 11, 24, 10, 4, 25, 14, 8, 26, 11, 12, 27, 15, 5, 28, 12, 9, 1, 29, 16, 13, 5, 30, 13, 6, 9, 31, 17, 10, 13, 32, 14, 14, 2
Offset: 1
Triangle begins (rows 1..28):
1;
2;
3;
4;
5;
6, 1;
7, 5;
8, 2;
9, 6;
10, 3;
11, 7;
12, 4;
13, 8;
14, 5;
15, 9, 1;
16, 6, 5;
17, 10, 9;
18, 7, 2;
19, 11, 6;
20, 8, 10;
21, 12, 3;
22, 9, 7;
23, 13, 11;
24, 10, 4;
25, 14, 8;
26, 11, 12;
27, 15, 5;
28, 12, 9, 1;
...
Figures A..H show the location (in the columns of the table) of the partitions of n = 1..8 (respectively) into consecutive parts that differ by 4:
. -----------------------------------------------------------
Fig: A B C D E F G H
. -----------------------------------------------------------
. n: 1 2 3 4 5 6 7 8
Row -----------------------------------------------------------
1 | [1];| 1; | 1; | 1; | 1; | 1; | 1; | 1; |
2 | | [2];| 2; | 2; | 2; | 2; | 2; | 2; |
3 | | | [3];| 3; | 3; | 3; | 3; | 3; |
4 | | | | [4];| 4; | 4; | 4; | 4; |
5 | | | | | [5];| 5; | 5; | 5; |
6 | | | | | | [6],[1];| 6, 1;| 6, 1; |
7 | | | | | | [5];| [7],5;| 7, 5; |
8 | | | | | | | | [8],[2];|
9 | | | | | | | | 9, [6];|
. -----------------------------------------------------------
Figure H: for n = 8 the partitions of 8 into consecutive parts that differ by 4 (but with the parts in increasing order) are [8] and [2, 6]. These partitions have one part and two parts respectively. On the other hand we can find the mentioned partitions in the columns 1 and 2 of this table, starting at the row 8.
.
Illustration of initial terms arranged into a triangular structure:
. _
. _|1|
. _|2 |
. _|3 |
. _|4 |
. _|5 _|
. _|6 |1|
. _|7 _|5|
. _|8 |2 |
. _|9 _|6 |
. _|10 |3 |
. _|11 _|7 |
. _|12 |4 |
. _|13 _|8 |
. _|14 |5 _|
. _|15 _|9 |1|
. _|16 |6 |5|
. _|17 _|10 _|9|
. _|18 |7 |2 |
. _|19 _|11 |6 |
. _|20 |8 _|10 |
. _|21 _|12 |3 |
. _|22 |9 |7 |
. |23 |13 |11 |
...
The number of horizontal line segments in the n-th row of the diagram equals A334461(n), the number of partitions of n into consecutive parts that differ by 4.
Tables of the same family where the consecutive parts differ by d are
A010766 (d=0),
A286001 (d=1),
A332266 (d=2),
A334945 (d=3), this sequence (d=4).
A334945
Irregular triangle read by rows: T(n,k), n >= 1, k >= 1, in which column k lists successive blocks of k consecutive integers that differ by 3, where the m-th block starts with m, m >= 1, and the first element of column k is in the row that is the k-th pentagonal number (A000326).
Original entry on oeis.org
1, 2, 3, 4, 5, 1, 6, 4, 7, 2, 8, 5, 9, 3, 10, 6, 11, 4, 12, 7, 1, 13, 5, 4, 14, 8, 7, 15, 6, 2, 16, 9, 5, 17, 7, 8, 18, 10, 3, 19, 8, 6, 20, 11, 9, 21, 9, 4, 22, 12, 7, 1, 23, 10, 10, 4, 24, 13, 5, 7, 25, 11, 8, 10, 26, 14, 11, 2, 27, 12, 6, 5, 28, 15, 9, 8, 29, 13, 12, 11, 30, 16, 7, 3
Offset: 1
Triangle begins:
1;
2;
3;
4;
5, 1;
6, 4;
7, 2;
8, 5;
9, 3;
10, 6;
11, 4;
12, 7, 1;
13, 5, 4;
14, 8, 7;
15, 6, 2;
16, 9, 5;
17, 7, 8;
18, 10, 3;
19, 8, 6;
20, 11, 9;
21, 9, 4;
22, 12, 7, 1;
...
Figures A..G show the location (in the columns of the table) of the partitions of n = 1..7 (respectively) into consecutive parts that differ by 3:
. -----------------------------------------------------
Fig: A B C D E F G
. -----------------------------------------------------
. n: 1 2 3 4 5 6 7
Row -----------------------------------------------------
1 | [1];| 1; | 1; | 1; | 1; | 1; | 1; |
2 | | [2];| 2; | 2; | 2; | 2; | 2; |
3 | | | [3];| 3; | 3; | 3; | 3; |
4 | | | | [4];| 4; | 4; | 4; |
5 | | | | | [5],[1];| 5, 1;| 5, 1; |
6 | | | | | 6, [4];| [6],4;| 6, 4; |
7 | | | | | | | [7],[2];|
8 | | | | | | | 8, [5];|
. -----------------------------------------------------
Figure G: for n = 7 the partitions of 7 into consecutive parts that differ by 3 (but with the parts in increasing order) are [7] and [2, 5]. These partitions have one part and two parts respectively. On the other hand we can find the mentioned partitions in the columns 1 and 2 of this table, starting at the row 7.
.
Illustration of initial terms arranged into a triangular structure:
. _
. _|1|
. _|2 |
. _|3 |
. _|4 _|
. _|5 |1|
. _|6 _|4|
. _|7 |2 |
. _|8 _|5 |
. _|9 |3 |
. _|10 _|6 |
. _|11 |4 _|
. _|12 _|7 |1|
. _|13 |5 |4|
. _|14 _|8 _|7|
. _|15 |6 |2 |
. _|16 _|9 |5 |
. _|17 |7 _|8 |
. _|18 _|10 |3 |
. _|19 |8 |6 |
. _|20 _|11 _|9 |
. _|21 |9 |4 _|
. |22 |12 |7 |1|
...
The number of horizontal line segments in the n-th row of the diagram equals A117277(n), the number of partitions of n into consecutive parts that differ by 3.
Tables of the same family where the consecutive parts differ by d are
A010766 (d=0),
A286001 (d=1),
A332266 (d=2), this sequence (d=3),
A334618(d=4).
A120885
Triangle read by rows where t(n,m) = ceiling(n/m).
Original entry on oeis.org
1, 2, 1, 3, 2, 1, 4, 2, 2, 1, 5, 3, 2, 2, 1, 6, 3, 2, 2, 2, 1, 7, 4, 3, 2, 2, 2, 1, 8, 4, 3, 2, 2, 2, 2, 1, 9, 5, 3, 3, 2, 2, 2, 2, 1, 10, 5, 4, 3, 2, 2, 2, 2, 2, 1, 11, 6, 4, 3, 3, 2, 2, 2, 2, 2, 1, 12, 6, 4, 3, 3, 2, 2, 2, 2, 2, 2, 1
Offset: 1
A123710
Indices k such that 4 = A123709(k) = number of nonzero terms in row k of triangle A123706.
Original entry on oeis.org
4, 6, 8, 9, 16, 25, 27, 32, 49, 64, 81, 121, 125, 128, 169, 243, 256, 289, 343, 361, 512, 529, 625, 729, 841, 961, 1024, 1331, 1369, 1681, 1849, 2048, 2187, 2197, 2209, 2401, 2809, 3125, 3481, 3721, 4096, 4489, 4913, 5041, 5329, 6241, 6561, 6859, 6889, 7921, 8192, 9409
Offset: 1
A174557
Triangle T(n, k) = -floor(n/k) with T(n, n) = 1, read by rows.
Original entry on oeis.org
1, -2, 1, -3, -1, 1, -4, -2, -1, 1, -5, -2, -1, -1, 1, -6, -3, -2, -1, -1, 1, -7, -3, -2, -1, -1, -1, 1, -8, -4, -2, -2, -1, -1, -1, 1, -9, -4, -3, -2, -1, -1, -1, -1, 1, -10, -5, -3, -2, -2, -1, -1, -1, -1, 1, -11, -5, -3, -2, -2, -1, -1, -1, -1, -1, 1, -12, -6, -4, -3, -2, -2, -1, -1, -1, -1, -1, 1
Offset: 1
Table begins:
1;
-2, 1;
-3, -1, 1;
-4, -2, -1, 1;
-5, -2, -1, -1, 1;
-6, -3, -2, -1, -1, 1;
-7, -3, -2, -1, -1, -1, 1;
-8, -4, -2, -2, -1, -1, -1, 1;
-9, -4, -3, -2, -1, -1, -1, -1, 1;
-10, -5, -3, -2, -2, -1, -1, -1, -1, 1;
-
[k eq n select 1 else -Floor(n/k): k in [1..n], n in [1..12]]; // G. C. Greubel, Mar 06 2021
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Table[If[k==n, 1, -Floor[n/k]], {n,12}, {k,n}]//Flatten (* G. C. Greubel, Mar 06 2021 *)
-
flatten([[1 if k==n else -(n//k) for k in [1..n]] for n in [1..12]]) # G. C. Greubel, Mar 06 2021
A278108
Irregular triangle read by rows: T(n,k) = floor(n/k^2) for 1 <= k^2 <= n.
Original entry on oeis.org
1, 2, 3, 4, 1, 5, 1, 6, 1, 7, 1, 8, 2, 9, 2, 1, 10, 2, 1, 11, 2, 1, 12, 3, 1, 13, 3, 1, 14, 3, 1, 15, 3, 1, 16, 4, 1, 1, 17, 4, 1, 1, 18, 4, 2, 1, 19, 4, 2, 1, 20, 5, 2, 1, 21, 5, 2, 1, 22, 5, 2, 1, 23, 5, 2, 1, 24, 6, 2, 1, 25, 6, 2, 1, 1, 26, 6, 2, 1, 1, 27, 6, 3, 1, 1, 28, 7, 3, 1, 1, 29, 7, 3, 1, 1
Offset: 1
The first 27 rows are:
1;
2;
3;
4, 1;
5, 1;
6, 1;
7, 1;
8, 2;
9, 2, 1;
10, 2, 1;
11, 2, 1;
12, 3, 1;
13, 3, 1;
14, 3, 1;
15, 3, 1;
16, 4, 1, 1;
17, 4, 1, 1;
18, 4, 2, 1;
19, 4, 2, 1;
20, 5, 2, 1;
21, 5, 2, 1;
22, 5, 2, 1;
23, 5, 2, 1;
24, 6, 2, 1;
25, 6, 2, 1, 1;
26, 6, 2, 1, 1;
27, 6, 3, 1, 1;
A033329
a(n) = floor(9/n).
Original entry on oeis.org
9, 4, 3, 2, 1, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 1
-
[Floor(9/n): n in [1..100]]; // Wesley Ivan Hurt, Apr 04 2023
-
Floor[9/Range[100]] (* or *) PadRight[{9,4,3,2,1,1,1,1,1},100,{0}](* Harvey P. Dale, Aug 05 2020 *)
A084934
Rectangular array T(m,n) (m>=1, n>=1) read by antidiagonals: row m consists of the numbers ( i + mj : i >= 0, j >= 0 ), sorted in increasing order, with repetitions allowed.
Original entry on oeis.org
0, 1, 0, 1, 1, 0, 2, 2, 1, 0, 2, 2, 2, 1, 0, 2, 3, 3, 2, 1, 0, 3, 3, 3, 3, 2, 1, 0, 3, 4, 4, 4, 3, 2, 1, 0, 3, 4, 4, 4, 4, 3, 2, 1, 0, 3, 4, 5, 5, 5, 4, 3, 2, 1, 0, 4, 5, 5, 5, 5, 5, 4, 3, 2, 1, 0, 4, 5, 6, 6, 6, 6, 5, 4, 3, 2, 1, 0, 4, 5, 6, 6, 6, 6, 6, 5, 4, 3, 2, 1, 0, 4, 6, 6, 7, 7, 7, 7, 6, 5, 4, 3, 2, 1, 0
Offset: 1
The northwest corner includes
m\n 1 2 3 4 5 6 7 8 9 10 ...
----------------------------
1 | 0 1 1 2 2 2 3 3 3 3 ...
2 | 0 1 2 2 3 3 4 4 4 5 ...
3 | 0 1 2 3 3 4 4 5 5 6 ...
4 | 0 1 2 3 4 4 5 5 6 6 ...
5 | 0 1 2 3 4 5 5 6 6 7 ...
Row m=0, for example, consists of the numbers i+j (i>=0, j>=0), sorted.
A128316
Triangle read by rows: A000012 * A128315 as infinite lower triangular matrices.
Original entry on oeis.org
1, 1, 1, 3, -1, 1, 2, 3, -2, 1, 4, -1, 4, -3, 1, 4, 3, -5, 7, -4, 1, 6, -3, 10, -13, 11, -5, 1, 4, 8, -14, 23, -24, 16, -6, 1, 7, -2, 15, -33, 46, -40, 22, -7, 1, 7, 4, -15, 47, -79, 86, -62, 29, -8, 1, 9, -6, 30, -73, 131, -166, 148, -91, 37, -9, 1, 7, 12, -37, 103, -204, 297, -314, 239, -128, 46, -10, 1
Offset: 1
First few rows of the triangle:
1;
1, 1;
3, -1, 1;
2, 3 -2, 1;
4, -1, 4, -3, 1;
4, 3, -5, 7, -4, 1;
6, -3, 10, -13, 11, -5, 1;
4, 8, -14, 23, -24, 16, -6, 1;
...
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A128316:= func< n,k | (&+[(-1)^(j+k)*Floor(n/j)*Binomial(j-1,k-1): j in [k..n]]) >;
[A128316(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jun 23 2024
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T[n_, k_]:= Sum[(-1)^(j+k)*Floor[n/j]*Binomial[j-1,k-1], {j,k,n}];
Table[T[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Jun 23 2024 *)
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def A128316(n,k): return sum((-1)^(j+k)*int(n//j)*binomial(j-1,k-1) for j in range(k,n+1))
flatten([[A128316(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jun 23 2024
a(28) = 1 inserted and more terms from
Georg Fischer, Jun 06 2023
Comments