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%I #27 Nov 29 2015 13:45:43
%S 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 128,665,243,146,73,37,18,8,256,2059,729,454,227,119,60,29,13,512,
%U 6305,2187,1394,697,373,192,97,47,21,1024,19171,6561,4246,2123,1151,600,311
%N Matrix defined by a(n,k) = 3^n*Fibonacci(k) - 2^n*Fibonacci(k-2), read by antidiagonals.
%C a(0,k) = A000045(k-1); a(1,k) = A000032(k); a(2,k) = A000285(k+1).
%C 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 Sum[a(n-k,k), {k,0,n}] = A098703(n+1), antidiagonal sums.
%C 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)|.
%H Michael De Vlieger, <a href="/A090888/b090888.txt">Table of n, a(n) for n = 0..10000</a>
%H Ross La Haye, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL12/LaHaye/lahaye5.html">Binary relations on the power set of an n-element set</a>, JIS 12 (2009) 09.2.6, table 4.
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>
%H Eric Weisstein, <a href="http://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>
%F a(n, k) = 3^n*Fibonacci(k) - 2^n*Fibonacci(k-2).
%F 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 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 O.g.f. (by rows) = (-2^n + (2^(n+1) - 3^n)x)/(-1+x+x^2). - _Ross La Haye_, Mar 30 2006
%F a(n,1) - a(n,0) = A003063(n+1). - _Ross La Haye_, Jun 22 2007
%F Binomial transform (by columns) of A118654. - _Ross La Haye_, Jun 22 2007
%e 1 0 1 1 2 3 5 8 13 21 34
%e 2 1 3 4 7 11 18 29 47 76 123
%e 4 5 9 14 23 37 60 97 157 254 411
%e 8 19 27 46 73 119 192 311 503 814 1317
%e 16 65 81 146 227 373 600 973 1573 2546 4119
%e 32 211 243 454 697 1151 1848 2999 4847 7846 12693
%e 64 665 729 1394 2123 3517 5640 9157 14797 23954 38751
%e a(5,3) = 454 because Fibonacci(3) = 2, Fibonacci(1) = 1 and (2 * 3^5) - (1 * 2^5) = 454.
%t Table[3^(n - k) Fibonacci@ k - 2^(n - k) Fibonacci[k - 2], {n, 0, 10}, {k, 0, n}] // Flatten (* _Michael De Vlieger_, Nov 28 2015 *)
%K nonn,tabl
%O 0,2
%A _Ross La Haye_, Feb 12 2004; revised Sep 24 2004, Sep 10 2005
%E More terms from _Ray Chandler_, Oct 27 2004
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