0,2

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).

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)|.

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]

Eric Weisstein, Fibonacci Number

Eric Weisstein, Lucas Number

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 (rlahaye(AT)new.rr.com), Mar 30 2006

a(n,1) - a(n,0) = A003063(n+1). - Ross La Haye (rlahaye(AT)new.rr.com), Jun 22 2007

Binomial transform (by columns) of A118654. - Ross La Haye (rlahaye(AT)new.rr.com), Jun 22 2007

{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}

a(5,3) = 454 because Fibonacci(3) = 2, Fibonacci(1) = 1 and (2 * 3^5) - (1 * 2^5) = 454.

Sequence in context: A153342 A144258 A056859 this_sequence A154794 A020781 A007432

Adjacent sequences: A090885 A090886 A090887 this_sequence A090889 A090890 A090891

nonn,tabl

Ross La Haye (rlahaye(AT)new.rr.com), Feb 12 2004; revised Sep 24 2004, Sep 10 2005.

More terms from Ray Chandler (rayjchandler(AT)sbcglobal.net), Oct 27 2004