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A180663 Mirror image of the Golden Triangle: T(n,k) = A001654(n-k) for n>=0 and 0<=k<=n. 2


%S 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,

%T 104,40,15,6,2,1,0,714,273,104,40,15,6,2,1,0,1870,714,273,104,40,15,6,

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

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

%C 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.

%C 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.

%H Reinhard Zumkeller, <a href="/A180663/b180663.txt">Rows n = 0..120 of triangle, flattened</a>

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

%e The first few rows of this triangle are:

%e 0;

%e 1, 0;

%e 2, 1, 0;

%e 6, 2, 1, 0;

%e 15, 6, 2, 1, 0;

%e 40, 15, 6, 2, 1, 0;

%p F:= combinat[fibonacci]:

%p T:= (n, k)-> F(n-k)*F(n-k+1):

%p seq(seq(T(n,k), k=0..n), n=0..10); # revised _Johannes W. Meijer_, Sep 13 2012

%o (Haskell)

%o a180663 n k = a180663_tabl !! n !! k

%o a180663_row n = a180663_tabl !! n

%o a180663_tabl = map reverse a180662_tabl

%o -- _Reinhard Zumkeller_, Jun 08 2013

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

%Y 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).

%K easy,nonn,tabl

%O 0,4

%A _Johannes W. Meijer_, Sep 21 2010

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Last modified February 24 16:12 EST 2020. Contains 332209 sequences. (Running on oeis4.)