%I A090888
%S A090888 1,2,0,4,1,1,8,5,3,1,16,19,9,4,2,32,65,27,14,7,3,64,211,81,46,23,11,5,
%T A090888 128,665,243,146,73,37,18,8,256,2059,729,454,227,119,60,29,13,512,6305,
%U A090888 2187,1394,697,373,192,97,47,21,1024,19171,6561,4246,2123,1151,600,311
%N A090888 Matrix defined by a(n,k) = 3^n(Fibonacci(k)) - 2^n(Fibonacci(k-2)), read 
               by antidiagonals.
%C A090888 a(0,k) = A000045(k-1); a(1,k) = A000032(k); a(2,k) = A000285(k+1).
%C A090888 a(n,1) = a(n-1,1) + a(n-1,3) for n > 0; a(n,1) = A001047(n) = 2^(2n) 
               - A083324(n); a(n,2) = A000244(n) = 2^(2n) - A005061(n); a(n,3) = 
               2a(n-1,4) for n > 0; a(n,3) = A027649(n); a(n,4) = A083313(n+1); 
               a(n,5) = A084171(n+1).
%C A090888 Sum[a(n-k,k), {k,0,n}] = A098703(n+1).
%C A090888 Let R, S and T be binary relations on the power set P(A) of a set A having 
               n = |A| elements such that for every element x, y of P(A), xRy if 
               x is a subset of y or y is a subset of x, xSy if x is a subset of 
               y and xTy if x is a proper subset of y. Then a(n,3) = |R|, a(n,2) 
               = |S| and a(n,1) = |T|. Note that a binary relation W on P(A) can 
               be defined also such that for every element x, y of P(A) xWy if x 
               is a proper subset of y and there are no z in P(A) such that x is 
               a proper subset of z and z is a proper subset of y. A090802(n,1) 
               = |W|. Also, a(n,0) = |P(A)|.
%D A090888 Ross La Haye, Binary Relations on the Power Set of an n-Element Set, 
               Journal of Integer Sequences, Vol. 12 (2009), Article 09.2.6. [From 
               Ross La Haye (rlahaye(AT)new.rr.com), Feb 22 2009]
%H A090888 Eric Weisstein, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">
               Fibonacci Number</a>
%H A090888 Eric Weisstein, <a href="http://mathworld.wolfram.com/LucasNumber.html">
               Lucas Number</a>
%F A090888 a(n, k) = 3^n(Fibonacci(k)) - 2^n(Fibonacci(k-2)).
%F A090888 a(n, 0) = 2^n, a(n, 1) = 3^n - 2^n, a(n, k) = a(n, k-1) + a(n, k-2) for 
               k > 1.
%F A090888 a(0, k) = Fibonacci(k-1), a(1, k) = Lucas(k), a(n, k) = 5a(n-1, k) - 
               6a(n-2, k) for n > 1.
%F A090888 O.g.f. (by rows) = (-2^n + (2^(n+1) - 3^n)x)/(-1+x+x^2). - Ross La Haye 
               (rlahaye(AT)new.rr.com), Mar 30 2006
%F A090888 a(n,1) - a(n,0) = A003063(n+1). - Ross La Haye (rlahaye(AT)new.rr.com), 
               Jun 22 2007
%F A090888 Binomial transform (by columns) of A118654. - Ross La Haye (rlahaye(AT)new.rr.com), 
               Jun 22 2007
%e A090888 {1}; {2,0}; {4,1,1}; {8,5,3,1}; {16,19,9,4,2}; {32,65,27,14,7,3};
%e A090888 {64,211,81,46,23,11,5}; {128,665,243,146,73,37,18,8}
%e A090888 a(5,3) = 454 because Fibonacci(3) = 2, Fibonacci(1) = 1 and (2 * 3^5) 
               - (1 * 2^5) = 454.
%Y A090888 Sequence in context: A153342 A144258 A056859 this_sequence A154794 A020781 
               A007432
%Y A090888 Adjacent sequences: A090885 A090886 A090887 this_sequence A090889 A090890 
               A090891
%K A090888 nonn,tabl
%O A090888 0,2
%A A090888 Ross La Haye (rlahaye(AT)new.rr.com), Feb 12 2004; revised Sep 24 2004, 
               Sep 10 2005.
%E A090888 More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Oct 27 
               2004