OFFSET
1,2
COMMENTS
Row sums = (1,5,14,34,74,...) = A094584. Alternating row sums = (1,1,4,4,12,12,...) given by F(m+1)-1 if m is even and F(m+2)-1 if m is odd. Central numbers = (1,5,16,47,...) = A094586.
Let p(n,x) = Sum_{k=0..n} F(k+1)*x^(n-k) and q(n,x) = x * q(n-1,x)+1, with q(0,x)=1. Then A094585 is the fission of sequence (p(n,x)) by sequence (q(n,x)); see A193842 for the definition of fission. A094585 is the mirror of A193999. [Clark Kimberling, Aug 11 2011]
LINKS
Muniru A Asiru, Rows n=1..150 of triangle, flattened
FORMULA
T(n, k) = F(n+3) - F(n+3-k) = F(n+1) + F(n) + ... + F(n+2-k), for k=1..n; n >= 1.
EXAMPLE
Rows 1 to 5:
1;
2, 3;
3, 5, 6;
5, 8, 10, 11;
8, 13, 16, 18, 19;
T(5,4) = F(8) - F(4) = 21 - 3 = 18;
T(5,4) = F(6) + F(5) + F(4) + F(3) = 8 + 5 + 3 + 2 = 18.
MATHEMATICA
See A193999.
Table[Fibonacci[n+3]-Fibonacci[n+3-k], {n, 1, 20}, {k, 1, n}]//TableForm (* Rigoberto Florez, Oct 03 2019 *)
PROG
(GAP) Flat(List([1..11], n->List([1..n], k->Fibonacci(n+3)-Fibonacci(n-k+3)))); # Muniru A Asiru, Apr 28 2019
KEYWORD
nonn,tabl
AUTHOR
Clark Kimberling, May 13 2004
STATUS
approved