A106408 - OEIS

A106408

Triangle, read by rows, where T(1,1) = 1; T(2,1) = T(2,2) = 2; for n > 2, T(n,n) = T(n-1,n-1) + T(n-2,n-2); T(n+1,n) = 2 * T(n,n); for all other entries, T(n,k) = T(n-1,k) + T(n-2,k).

1, 2, 2, 3, 4, 3, 5, 6, 6, 5, 8, 10, 9, 10, 8, 13, 16, 15, 15, 16, 13, 21, 26, 24, 25, 24, 26, 21, 34, 42, 39, 40, 40, 39, 42, 34, 55, 68, 63, 65, 64, 65, 63, 68, 55, 89, 110, 102, 105, 104, 104, 105, 102, 110, 89, 144, 178, 165, 170, 168, 169, 168, 170, 165, 178, 144

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OFFSET

1,2

COMMENTS

Row sums are A004798 (convolution of Fibonacci numbers 1,2,3,5,... with themselves). Central numbers of the rows are A006498 (a(n) = a(n-1)+a(n-3)+a(n-4)). First column and main diagonal are Fibonacci numbers 1,2,3,5,... First subdiagonal are 2*Fibonacci numbers. T(n,k) = F(n-k+2)*F(k+1) where F(m) is the m-th Fibonacci number. For the antidiagonal sums b(n): b(1) = 1, b(2) = 2, then b(n) = b(n-1) + b(n-2) + F(floor((n+3)/2)).

T(n,k) is the number of Boolean intervals of the form [s_k,w] in the weak order on S_n, for a fixed simple reflection s_k. - Bridget Tenner, Jan 16 2020

LINKS

Table of n, a(n) for n=1..66.

B. E. Tenner, Interval structures in the Bruhat and weak orders, arXiv:2001.05011 [math.CO], 2020.

FORMULA

G.f.: (1+x+y+x*y)/((1-x-x^2)*(1-y-y^2)) [U coordinates] - N. J. A. Sloane, Jun 01 2005

EXAMPLE

Triangle begins

2, 2;

3, 4, 3;

5, 6, 6, 5;

8, 10, 9, 10, 8;

CROSSREFS