|
| |
|
|
A143187
|
|
Triangle read by rows: T(n, k) = f(k) for 1 <= k <= floor(n/2), T(n, k) = f(n-k) for floor(n/2) < k <= n-1, with T(n, 0) = 1, T(n, n) = 1, and f(k) = (1/2)*(3 - (-1)^k)*k.
|
|
2
|
|
|
|
1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 1, 2, 2, 6, 2, 2, 1, 1, 2, 2, 6, 6, 2, 2, 1, 1, 2, 2, 6, 4, 6, 2, 2, 1, 1, 2, 2, 6, 4, 4, 6, 2, 2, 1, 1, 2, 2, 6, 4, 10, 4, 6, 2, 2, 1, 1, 2, 2, 6, 4, 10, 10, 4, 6, 2, 2, 1, 1, 2, 2, 6, 4, 10, 6, 10, 4, 6, 2, 2, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,5
|
|
|
LINKS
|
|
|
|
FORMULA
|
T(n, k) = f(k) for 1 <= k <= floor(n/2), T(n, k) = f(n-k) for floor(n/2) < k <= n-1, with T(n, 0) = 1, T(n, n) = 1, and f(k) = (1/2)*(3 - (-1)^k)*k.
T(n, n-k) = T(n, k).
Sum_{k=0..n} T(n, k) = (1/16)*(33 + 3*(-1)^n - 4*cos(n*Pi/2) - 4*sin(n*Pi/2)*n + 6*n^2) - [n=0] (row sums). - G. C. Greubel, Apr 30 2024
|
|
|
EXAMPLE
|
Triangle begins as:
1;
1, 1;
1, 2, 1;
1, 2, 2, 1;
1, 2, 2, 2, 1;
1, 2, 2, 2, 2, 1;
1, 2, 2, 6, 2, 2, 1;
1, 2, 2, 6, 6, 2, 2, 1;
1, 2, 2, 6, 4, 6, 2, 2, 1;
1, 2, 2, 6, 4, 4, 6, 2, 2, 1;
1, 2, 2, 6, 4, 10, 4, 6, 2, 2, 1;
|
|
|
MATHEMATICA
|
f[n_]= (3-(-1)^n)*n/2;
T[n_, k_]:= If[k*(n-k)==0, 1, If[k <= Floor[n/2], f[k], f[n-k]]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten
|
|
|
PROG
|
(Magma)
f:= func< n | (3-(-1)^n)*n/2 >;
A143187:= func< n, k | k eq 0 or k eq n select 1 else k le Floor(n/2) select f(k) else f(n-k) >;
(SageMath)
def f(n): return (3-(-1)^n)*n/2
if k==0 or k==n: return 1
elif k<=n//2: return f(k)
else: return f(n-k)
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
