OFFSET
1,3
COMMENTS
The number of terms in row n is 3*n-1 = A016789(n-1).
Fibonacci(n) = T(n+k,n) - T(n+k-1,n) for n >= 1, k = 1,2,3,...
The elements b(k) of the main diagonal, superdiagonal 1 and all subdiagonals have the recursive formula: b(k) = 2*b(k-1) + b(k-2) - 2*b(k-3) - b(k-4) for k > 4.
FORMULA
T(n,k) = T(n,3*k-n) for 1 <= k <= 3*n-1.
T(n,k) = Sum_{u=2*(n-k)+3..2*n-k+1} ceiling(u/2)*A065941(k-2,u-2*(n-k)-3) for n >= 3, 3 <= k <= n.
T(n,k) = Sum_{m1=1..k-n} A208354(m1)*binomial(n-m1-1, k-n-m1) + Sum_{m2=1..2*n-k} A208354(m2)*binomial(n-m2-1, 2*n-k-m2) for n >= 2, n+1 <= k <= 2*n-1.
T(n,k) = Sum_{u=2*(k-2*n)+3..k-n+1} ceiling(u/2)*A065941(3*n-k-2,u-2*(k-2*n)-3) for n>= 3, 2*n <= k <= 3*(n-1).
T(n,k) = A208354(k) + (n-k)*Fibonacci(k) for n >= 3, 3 <= k <= n.
T(n,k) = A029907(k-1) + (n-k+1)*Fibonacci(k) for n >= 2, 3 <= k <= n+1.
EXAMPLE
Triangle begins:
n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20...
1 1 1
2 2 2 2 2 2
3 3 3 4 4 4 4 3 3
4 4 4 6 7 8 8 8 7 6 4 4
5 5 5 8 10 13 15 16 16 15 13 10 8 5 5
6 6 6 10 13 18 23 28 31 32 31 28 23 18 13 10 6 6
7 7 7 12 16 23 31 41 51 59 63 63 59 51 41 31 23 16 12 7 7
...
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Lechoslaw Ratajczak, Jul 04 2020
STATUS
approved