A180663 Mirror image of the Golden Triangle: T(n,k) = A001654(n-k) for n>=0 and 0<=k<=n.

0, 1, 0, 2, 1, 0, 6, 2, 1, 0, 15, 6, 2, 1, 0, 40, 15, 6, 2, 1, 0, 104, 40, 15, 6, 2, 1, 0, 273, 104, 40, 15, 6, 2, 1, 0, 714, 273, 104, 40, 15, 6, 2, 1, 0, 1870, 714, 273, 104, 40, 15, 6, 2, 1, 0, 4895, 1870, 714, 273, 104, 40, 15, 6, 2, 1, 0

Offset: 0

Author

Johannes W. Meijer, Sep 21 2010

Keywords

• easy
• nonn
• tabl

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.

Examples

			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;
		

Links

• Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Crossrefs

Cf. A180662 (Golden Triangle), A001654 (Golden Rectangle numbers), A000045 (F(n)).

The triangle sums lead to: A064831 (Row1, Kn21, Kn22, Kn3, Ca2, Ca3, Gi2, Gi3), A077916 (Row2), A180664 (Kn23), A180665 (Kn11, Kn12, Kn13, Fi1, Ze1), A180665(2*n) (Kn4, Fi2, Ze4), A115730(n+1) (Ca1, Ze3), A115730(3*n+1) (Ca4, Ze2), A180666 (Gi1), A180666(4*n) (Gi4).

Programs

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

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 *)

Formula

T(n,k) = F(n-k)*F(n-k+1) with F(n) = A000045(n), for n>=0 and 0<=k<=n.