OFFSET
0,4
COMMENTS
This triangle is the mirror image of the Golden Triangle A180662. The terms in the n-th row of the triangle are the first (n+1) golden rectangle numbers in reversed order. The golden rectangle numbers are A001654(n)=F(n)*F(n+1), with F(n) the Fibonacci numbers.
The chess sums, see A180662 for their definitions, mirror those of the Golden Triangle: Row1 & Row1; Row 2 & Row2; Kn1 and Kn2; Kn3 and Kn4; Fi1 and Fi2; Ca1 and Ca2; Ca3 and Ca4; Gi1 and Gi2; Gi3 and Gi4; Ze1 and Ze2; Ze3 and Ze4.
LINKS
Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened
FORMULA
T(n,k) = F(n-k)*F(n-k+1) with F(n) = A000045(n), for n>=0 and 0<=k<=n.
EXAMPLE
The first few rows of this triangle are:
0;
1, 0;
2, 1, 0;
6, 2, 1, 0;
15, 6, 2, 1, 0;
40, 15, 6, 2, 1, 0;
MAPLE
F:= combinat[fibonacci]:
T:= (n, k)-> F(n-k)*F(n-k+1):
seq(seq(T(n, k), k=0..n), n=0..10); # revised Johannes W. Meijer, Sep 13 2012
MATHEMATICA
Module[{nn=20, fb}, fb=Times@@@Partition[Fibonacci[Range[0, (nn(nn+1))/2]], 2, 1]; Table[ Reverse[Take[fb, n]], {n, nn}]]//Flatten (* Harvey P. Dale, Jan 30 2023 *)
PROG
(Haskell)
a180663 n k = a180663_tabl !! n !! k
a180663_row n = a180663_tabl !! n
a180663_tabl = map reverse a180662_tabl
-- Reinhard Zumkeller, Jun 08 2013
CROSSREFS
KEYWORD
AUTHOR
Johannes W. Meijer, Sep 21 2010
STATUS
approved