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 A090888 Matrix defined by a(n,k) = 3^n*Fibonacci(k) - 2^n*Fibonacci(k-2), read by antidiagonals. 11
 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, 128, 665, 243, 146, 73, 37, 18, 8, 256, 2059, 729, 454, 227, 119, 60, 29, 13, 512, 6305, 2187, 1394, 697, 373, 192, 97, 47, 21, 1024, 19171, 6561, 4246, 2123, 1151, 600, 311 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS a(0,k) = A000045(k-1); a(1,k) = A000032(k); a(2,k) = A000285(k+1). 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). Sum[a(n-k,k), {k,0,n}] = A098703(n+1), antidiagonal sums. 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)|. LINKS Michael De Vlieger, Table of n, a(n) for n = 0..10000 Ross La Haye, Binary relations on the power set of an n-element set, JIS 12 (2009) 09.2.6, table 4. Eric Weisstein, Fibonacci Number Eric Weisstein, Lucas Number FORMULA a(n, k) = 3^n*Fibonacci(k) - 2^n*Fibonacci(k-2). 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. 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. O.g.f. (by rows) = (-2^n + (2^(n+1) - 3^n)x)/(-1+x+x^2). - Ross La Haye, Mar 30 2006 a(n,1) - a(n,0) = A003063(n+1). - Ross La Haye, Jun 22 2007 Binomial transform (by columns) of A118654. - Ross La Haye, Jun 22 2007 EXAMPLE 1    0    1    1    2    3    5    8    13    21    34    2    1    3    4    7   11   18   29    47    76   123    4    5    9   14   23   37   60   97   157   254   411    8   19   27   46   73  119  192  311   503   814  1317   16   65   81  146  227  373  600  973  1573  2546  4119   32  211  243  454  697 1151 1848 2999  4847  7846 12693   64  665  729 1394 2123 3517 5640 9157 14797 23954 38751 a(5,3) = 454 because Fibonacci(3) = 2, Fibonacci(1) = 1 and (2 * 3^5) - (1 * 2^5) = 454. MATHEMATICA 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 *) CROSSREFS Sequence in context: A188448 A304789 A264379 * A154794 A177264 A326758 Adjacent sequences:  A090885 A090886 A090887 * A090889 A090890 A090891 KEYWORD nonn,tabl AUTHOR Ross La Haye, Feb 12 2004; revised Sep 24 2004, Sep 10 2005 EXTENSIONS More terms from Ray Chandler, Oct 27 2004 STATUS approved

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Last modified April 22 18:24 EDT 2021. Contains 343177 sequences. (Running on oeis4.)