OFFSET
1,2
COMMENTS
The sum of row n equals A138879(n), the sum of all parts in the last section of the set of partitions of n.
T(n,k) is also the number of cubic cells (or cubes) added at the n-th stage in the k-th level starting from the base in the tower described in A221529, assuming that the tower is an object under construction (see the example). - Omar E. Pol, Jan 20 2022
LINKS
EXAMPLE
Triangle begins:
1;
3;
4, 1;
7, 3, 1;
6, 4, 3, 1, 1;
12, 7, 4, 3, 3, 1, 1;
8, 6, 7, 4, 4, 3, 3, 1, 1, 1, 1;
15, 12, 6, 7, 7, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1;
13, 8, 12, 6, 6, 7, 7, 4, 4, 4, 4, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1;
...
From Omar E. Pol, Jan 13 2022: (Start)
Illustration of the first six rows of triangle showing the growth of the symmetric tower described in A221529:
Level k: 1 2 3 4 5 6 7
Stage
n _ _ _ _ _ _ _ _
| _ |
1 | |_| |
|_ _ _ _ _ _ _ _|
| _ |
| | |_ |
2 | |_ _| |
|_ _ _ _ _ _ _ _|_ _ _ _ _ _
| _ | _ |
| | | | |_| |
3 | |_|_ _ | |
| |_ _| | |
|_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _
| _ | _ | _ |
| | | | | |_ | |_| |
4 | | |_ | |_ _| | |
| |_ |_ _ | | |
| |_ _ _| | | |
|_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _|_ _ _ _ _ _ _ _
| _ | _ | _ | _ | _ |
| | | | | | | | |_ | |_| | |_| |
| | | | |_|_ _ | |_ _| | | |
5 | |_|_ | |_ _| | | | |
| |_ _ _ | | | | |
| |_ _ _| | | | | |
|_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _|_ _ _ _|_ _ _ _|_ _ _ _ _ _
| _ | _ | _ | _ | _ | _ | _ |
| | | | | | | | | | | |_ | | |_ | |_| | |_| |
| | | | | |_ | |_|_ _ | |_ _| | |_ _| | | |
| | |_ _ | |_ |_ _ | |_ _| | | | | |
6 | |_ | | |_ _ _| | | | | | |
| |_ |_ _ _ | | | | | | |
| |_ _ _ _| | | | | | | |
|_ _ _ _ _ _ _ _|_ _ _ _ _ _|_ _ _ _ _|_ _ _ _|_ _ _ _|_ _ _|_ _ _|
.
Every cell in the diagram of the symmetric representation of sigma represents a cubic cell or cube.
For n = 6 and k = 3 we add four cubes at 6th stage in the third level of the structure of the tower starting from the base so T(6,3) = 4.
For n = 9 another connection with the tower is as follows:
First we take the columns from the above triangle and build a new triangle in which all columns start at row 1 as shown below:
.
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;
3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3;
4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4;
7, 7, 7, 7, 7, 7, 7;
6, 6, 6, 6, 6;
12, 12, 12;
8, 8;
15;
13;
.
Then we rotate the triangle by 90 degrees as shown below:
_
1; | |
1; | |
1; | |
1; | |
1; | |
1; | |
1; |_|_
1, 3; | |
1, 3; | |
1, 3; | |
1, 3; |_ _|_
1, 3, 4; | | |
1, 3, 4; | | |
1, 3, 4; | | |
1, 3, 4; |_ _|_|_
1, 3, 4, 7; | | |
1, 3, 4, 7; |_ _ _| |_
1, 3, 4, 7, 6; | | |
1, 3, 4, 7, 6; |_ _ _|_ _|_
1, 3, 4, 7, 6, 12; |_ _ _ _| | |_
1, 3, 4, 7, 6, 12, 8; |_ _ _ _|_|_ _|_ _
1, 3, 4, 7, 6, 12, 8, 15; 13; |_ _ _ _ _|_ _|_ _|
.
Lateral view
of the tower
. _ _ _ _ _ _ _ _ _
|_| | | | | | | |
|_ _|_| | | | | |
|_ _| _|_| | | |
|_ _ _| _|_| |
|_ _ _| _| _ _|
|_ _ _ _| |
|_ _ _ _| _ _|
| |
|_ _ _ _ _|
.
Top view
of the tower
.
The sum of the m-th row of the new triangle equals A024916(j) where j is the length of the m-th row, equaling the number of cubic cells in the m-th level of the tower. For example: the last row of triangle has 9 terms and the sum of the last row is 1 + 3 + 4 + 7 + 6 + 12 + 8 + 15 + 13 = A024916(9) = 69, equaling the number of cubes in the base of the tower. (End)
MATHEMATICA
A339278[rowmax_]:=Table[Flatten[Table[ConstantArray[DivisorSigma[1, n-m], PartitionsP[m]-PartitionsP[m-1]], {m, 0, n-1}]], {n, rowmax}];
A339278[15] (* Generates 15 rows *) (* Paolo Xausa, Feb 17 2023 *)
PROG
(PARI) f(n) = numbpart(n-1);
T(n, k) = {if (k > f(n), error("invalid k")); if (k==1, return (sigma(n))); my(s=0); while (k <= f(n-1), s++; n--; ); sigma(1+s); }
tabf(nn) = {for (n=1, nn, for (k=1, f(n), print1(T(n, k), ", "); ); print; ); } \\ Michel Marcus, Jan 13 2021
(PARI) A339278(rowmax)=vector(rowmax, n, concat(vector(n, m, vector(numbpart(m-1)-numbpart(m-2), i, sigma(n-m+1)))));
A339278(15) \\ Generates 15 rows \\ Paolo Xausa, Feb 17 2023
CROSSREFS
Sum of divisors of A336811.
Row n has length A000041(n-1).
Every column gives A000203.
The length of the m-th block in row n is A187219(m), m >= 1.
Row sums give A138879.
Cf. A337209 (another version).
KEYWORD
nonn,tabf
AUTHOR
Omar E. Pol, Nov 29 2020
STATUS
approved