|
|
A136157
|
|
Triangle by columns, (3, 1, 0, 0, 0, ...) in every column.
|
|
2
|
|
|
3, 1, 3, 0, 1, 3, 0, 0, 1, 3, 0, 0, 0, 1, 3, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 3
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,1
|
|
COMMENTS
|
Infinite lower triangular matrix with (3, 3, 3, ...) in the main diagonal and (1, 1, 1, ...) in the subdiagonal, with the rest zeros.
|
|
LINKS
|
|
|
FORMULA
|
T(n, k) = 3 if k = n, T(n, k) = 1 if k = n-1, otherwise T(n, k) = 0.
T(n, k) = 2 + (-1)^(n+k) for k >= n-1, otherwise T(n, k) = 0.
Sum_{k=0..n} T(n, k) = 4 - [n=0].
Sum_{k=0..n} (-1)^k*T(n, k) = (-2)^n + [n=0].
Sum_{k=0..floor(n/2)} T(n-k, k) = 2 + (-1)^n.
Sum_{k=0..floor(n/2)} (-1)^k*T(n-k, k) = (2 + (-1)^n)*(-1)^floor(n/2). (End)
|
|
EXAMPLE
|
First few rows of the triangle:
3;
1, 3;
0, 1, 3;
0, 0, 1, 3;
0, 0, 0, 1, 3;
0, 0, 0, 0, 1, 3;
...
|
|
MATHEMATICA
|
Table[PadLeft[{1, 3}, n, {0}], {n, 0, 20}]//Flatten (* Harvey P. Dale, Apr 04 2018 *)
|
|
PROG
|
(Magma)
if k gt n-2 then return 2 + (-1)^(n+k);
else return 0;
end if;
end function;
[T(n, k): k in [0..n], n in [0..12]]; // G. C. Greubel, Dec 26 2023
(SageMath)
if k>n-2: return 2 + (-1)^(n+k)
else: return 0
flatten([[T(n, k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Dec 26 2023
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|