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A090888
Matrix defined by a(n,k) = 3^n*Fibonacci(k) - 2^n*Fibonacci(k-2), read by antidiagonals.
12
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
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
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: A355917 A304789 A264379 * A154794 A177264 A326758
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