

A106408


Triangle, read by rows, where T(1,1) = 1; T(2,1) = T(2,2) = 2; for n > 2, T(n,n) = T(n1,n1) + T(n2,n2); T(n+1,n) = 2 * T(n,n); for all other entries, T(n,k) = T(n1,k) + T(n2,k).


1



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(n1)+a(n3)+a(n4)). First column and main diagonal are Fibonacci numbers 1,2,3,5,... First subdiagonal are 2*Fibonacci numbers. T(n,k) = F(nk+2)*F(k+1) where F(m) is the mth Fibonacci number. For the antidiagonal sums b(n): b(1) = 1, b(2) = 2, then b(n) = b(n1) + b(n2) + 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)/((1xx^2)*(1yy^2)) [U coordinates]  N. J. A. Sloane, Jun 01 2005


EXAMPLE

Triangle begins
1;
2, 2;
3, 4, 3;
5, 6, 6, 5;
8, 10, 9, 10, 8;


CROSSREFS

Cf. A000045, A004798, A006498.
Sequence in context: A128282 A146985 A132993 * A143061 A096858 A037254
Adjacent sequences: A106405 A106406 A106407 * A106409 A106410 A106411


KEYWORD

nonn,tabl


AUTHOR

Gerald McGarvey, May 28 2005


STATUS

approved



