A090888 Matrix defined by a(n,k) = 3^n*Fibonacci(k) - 2^n*Fibonacci(k-2), read by antidiagonals.

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

Author

Ross La Haye, Feb 12 2004; revised Sep 24 2004, Sep 10 2005

Keywords

• nonn
• tabl

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)|.

Examples

			   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.
		

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

Programs

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 *)

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

Extensions

More terms from Ray Chandler, Oct 27 2004