A047790 a(n) = Fibonacci(2*n)-2^n+1.

0, 0, 0, 1, 6, 24, 81, 250, 732, 2073, 5742, 15664, 42273, 113202, 301428, 799273, 2112774, 5571816, 14668209, 38563882, 101285580, 265817145, 697214430, 1827923296, 4790749761, 12552714594, 32884171236, 86133353545, 225582998262, 590749858968, 1546935014097

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• T. Mansour and A. O. Munagi, Block-connected set partitions, European J. Combin., 31 (2010), 887-902, Table 3 column 3.
• Index entries for linear recurrences with constant coefficients, signature (6,-12,9,-2).

Programs

Maple

with(combstruct): SeqSeqSeqL := [T, {T=Sequence(S, card > 1), S=Sequence(U, card > 1), U=Sequence(Z, card >0)}, unlabeled]: seq(count(SeqSeqSeqL, size=j+1), j=0..29); # Zerinvary Lajos, Apr 16 2009

Mathematica

Table[Fibonacci[2n]-2^n+1,{n,0,30}] (* or *) LinearRecurrence[ {6,-12,9,-2},{0,0,0,1},30] (* Harvey P. Dale, Jan 19 2012 *)
CoefficientList[Series[x^3/(1 - 6 x + 12 x^2 - 9 x^3 + 2 x^4), {x, 0, 30}], x] (* Benedict W. J. Irwin, Nov 02 2016 *)

PARI

concat(vector(3), Vec(x^3/(1-6*x+12*x^2-9*x^3+2*x^4) + O(x^40))) \\ Colin Barker, Nov 02 2016

Sage

[lucas_number1(n,3,1)-lucas_number1(n,3,2) for n in range(0,30)] # Zerinvary Lajos, Jul 06 2008

Formula

a(0)=0, a(1)=0, a(2)=0, a(3)=1, a(n)=6*a(n-1)-12*a(n-2)+ 9*a(n-3)- 2*a(n-4). - Harvey P. Dale, Jan 19 2012

G.f.: x^3/(1 - 6*x + 12*x^2 - 9*x^3 + 2*x^4). - Benedict W. J. Irwin, Nov 02 2016

a(n) = (1-2^n-((3-sqrt(5))/2)^n/sqrt(5)+((3+sqrt(5))/2)^n/sqrt(5)). - Colin Barker, Nov 02 2016