|
|
A350637
|
|
Triangle read by rows: T(n,k) in which row n lists the first n terms of A024916 in reverse order, 1 <= k <= n.
|
|
1
|
|
|
1, 4, 1, 8, 4, 1, 15, 8, 4, 1, 21, 15, 8, 4, 1, 33, 21, 15, 8, 4, 1, 41, 33, 21, 15, 8, 4, 1, 56, 41, 33, 21, 15, 8, 4, 1, 69, 56, 41, 33, 21, 15, 8, 4, 1, 87, 69, 56, 41, 33, 21, 15, 8, 4, 1, 99, 87, 69, 56, 41, 33, 21, 15, 8, 4, 1, 127, 99, 87, 69, 56, 41, 33, 21, 15, 8, 4, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
T(n,k) is the number of cubic cells (or cubes) in the k-th level starting from the base of the stepped pyramid with n levels described in A245092 (see example).
|
|
LINKS
|
|
|
FORMULA
|
T(n,k) = Sum_{j=1..n} A272172(j,k).
|
|
EXAMPLE
|
Triangle begins:
1;
4, 1;
8, 4, 1;
15, 8, 4, 1;
21, 15, 8, 4, 1;
33, 21, 15, 8, 4, 1;
41, 33, 21, 15, 8, 4, 1;
56, 41, 33, 21, 15, 8, 4, 1;
69, 56, 41, 33, 21, 15, 8, 4, 1;
87, 69, 56, 41, 33, 21, 15, 8, 4, 1;
99, 87, 69, 56, 41, 33, 21, 15, 8, 4, 1;
127, 99, 87, 69, 56, 41, 33, 21, 15, 8, 4, 1;
...
For n = 9 the lateral view and top view of the stepped pyramid described in A245092 look as shown below:
_
9 1 |_|_
8 4 |_ _|_
7 8 |_ _|_|_
6 15 |_ _ _| |_
5 21 |_ _ _|_ _|_
4 33 |_ _ _ _| | |_
3 41 |_ _ _ _|_|_ _|_
2 56 |_ _ _ _ _|_|_ |_
1 69 |_ _ _ _ _|_ _|_ _|
.
Level Row 9 Lateral view of
k T(9,k) the stepped pyramid
.
_ _ _ _ _ _ _ _ _
|_| | | | | | | | |
|_ _|_| | | | | | |
|_ _| _|_| | | | |
|_ _ _| _|_| | |
|_ _ _| _| _ _|_|
|_ _ _ _| _| |
|_ _ _ _| |_ _|
|_ _ _ _ _|
|_ _ _ _ _|
.
Top view of
the stepped pyramid
.
For n = 9 and k = 1 there are 69 cubic cells in the level 1 starting from the base of the stepped pyramid, so T(9,1) = 69.
For n = 9 and k = 9 there is only one cubic cell in the level k = 9 (the top) of the stepped pyramid, so T(9,9) = 1.
The volume of the stepped pyramid (also the total number of cubic cells) represents the 9th term of the convolution of A000203 and A000027 hence it's equal to A175254(9) = 248, equaling the sum of the 9th row of triangle.
|
|
MATHEMATICA
|
|
|
PROG
|
(PARI) row(n) = Vecrev(vector(n, k, sum(i=1, k, k\i*i))); \\ Michel Marcus, Jan 22 2022
|
|
CROSSREFS
|
Column k gives A024916 starting in row k.
Cf. A000027, A000203, A004736, A196020, A235791, A236104, A237591, A237593, A245092, A262626, A272172.
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|