OFFSET
0,3
COMMENTS
If the rows of the triangle are extended for k > n, the array A144216 is created, which is symmetrical to the main diagonal and therefore contains no new information compared to this triangle.
FORMULA
Recurrence: T(n, n) = n + T(n, n-1) starting with T(0, 0) = 0.
For k <> n: T(n, k) = n + T(n-1, k).
T(n, k) = t(n) + t(k), where t(n) are the triangular numbers A000217.
G.f.: (x + x*(2 - 5*x + x^2)*y + x^4*y^2)/((1 - x)^3*(1 - x*y)^3). - Stefano Spezia, Dec 07 2023
EXAMPLE
Triangle T(n, k) starts:
0 | 0;
1 | 1, 2;
2 | 3, 4, 6;
3 | 6, 7, 9, 12;
4 | 10, 11, 13, 16, 20;
5 | 15, 16, 18, 21, 25, 30;
6 | 21, 22, 24, 27, 31, 36, 42;
7 | 28, 29, 31, 34, 38, 43, 49, 56;
8 | 36, 37, 39, 42, 46, 51, 57, 64, 72;
9 | 45, 46, 48, 51, 55, 60, 66, 73, 81, 90;
10 | 55, 56, 58, 61, 65, 70, 76, 83, 91, 100, 110;
.
Start at row 0, column 0 with 0. Go down by adding the column index in step n. At row n, restart the counting and go n steps right by adding the row index in step n, then change direction and go down again by adding the column index. After 3*n steps on this path you are at T(2*n, n) which is 2*triangular(n) + (triangular(2*n) - triangular(n)) = (5*n^2 + 3*n)/2. These are the sliced heptagonal numbers A147875 (see the illustration of Leo Tavares).
.
The equation T(n, k) = (n*(n + 1) + k*(k + 1))/2 can be extended to all n, k in ZZ.
[n\k] ... -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 ...
-------------------------------------------------------------
[-5] ..., 25, 20, 16, 13, 11, 10, 10, 11, 13, 16, 20, 25, ...
[-4] ..., 21, 16, 12, 9, 7, 6, 6, 7, 9, 12, 16, 21, ...
[-3] ..., 18, 13, 9, 6, 4, 3, 3, 4, 6, 9, 13, 18, ...
[-2] ..., 16, 11, 7, 4, 2, 1, 1, 2, 4, 7, 11, 16, ...
[-1] ..., 15, 10, 6, 3, 1, 0, 0, 1, 3, 6, 10, 15, ...
[ 0] ..., 15, 10, 6, 3, 1, 0, 0, 1, 3, 6, 10, 15, ...
[ 1] ..., 16, 11, 7, 4, 2, 1, 1, 2, 4, 7, 11, 16, ...
[ 2] ..., 18, 13, 9, 6, 4, 3, 3, 4, 6, 9, 13, 18, ...
[ 3] ..., 21, 16, 12, 9, 7, 6, 6, 7, 9, 12, 16, 21, ...
[ 4] ..., 25, 20, 16, 13, 11, 10, 10, 11, 13, 16, 20, 25, ...
MAPLE
T := (n, k) -> (n*(n + 1) + k*(k + 1)) / 2:
for n from 0 to 10 do seq(T(n, k), k = 0..n) od;
MATHEMATICA
Module[{n=1}, NestList[Append[#+n, n*++n]&, {0}, 10]] (* or *)
Table[(n(n+1)+k(k+1))/2, {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Dec 07 2023 *)
PROG
(Python) # A purely additive construction:
from functools import cache
@cache
def a_row(n: int) -> list[int]:
if n == 0: return [0]
row = a_row(n - 1) + [0]
for k in range(n): row[k] += n
row[n] = row[n - 1] + n
return row
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Dec 07 2023
STATUS
approved