|
|
A055588
|
|
a(n) = 3*a(n-1) - a(n-2) - 1 with a(0) = 1 and a(1) = 2.
|
|
18
|
|
|
1, 2, 4, 9, 22, 56, 145, 378, 988, 2585, 6766, 17712, 46369, 121394, 317812, 832041, 2178310, 5702888, 14930353, 39088170, 102334156, 267914297, 701408734, 1836311904, 4807526977, 12586269026, 32951280100, 86267571273
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Number of directed column-convex polyominoes with area n+2 and having two cells in the bottom row. - Emeric Deutsch, Jun 14 2001
a(n) is the length of the list generated by the substitution: 3->3, 4->(3,4,6), 6->(3,4,6,6): {3, 4}, {3, 3, 4, 6}, {3, 3, 3, 4, 6, 3, 4, 6, 6}, {3, 3, 3, 3, 4, 6, 3, 4, 6, 6, 3, 3, 4, 6, 3, 4, 6, 6, 3, 4, 6, 6}, etc. - Wouter Meeussen, Nov 23 2003
|
|
LINKS
|
|
|
FORMULA
|
a(n) = (((3 + sqrt(5))/2)^n - ((3 - sqrt(5))/2)^n)/sqrt(5) + 1.
G.f.: (1 - 2*x)/((1 - 3*x + x^2)*(1-x)).
a(n) = 4*a(n-1) - 4*a(n-2) + a(n-3);
a(n) = Sum_{k=0..floor(n/3)} binomial(n-k, 2*k)2^(n-3*k). (End)
a(n) = Sum_{k=0..n} Fibonacci(2*k+2)*(2*0^(n-k) - 1).
a(n) = Sum_{k=0..2*n+1} ((-1)^(k+1))*Fibonacci(k). - Michel Lagneau, Feb 03 2014
|
|
MAPLE
|
g:=z/(1-3*z+z^2): gser:=series(g, z=0, 43): seq(abs(coeff(gser, z, n)+1), n=0..27); # Zerinvary Lajos, Mar 22 2009
|
|
MATHEMATICA
|
Table[Fibonacci[2n] +1, {n, 0, 40}] (* or *) LinearRecurrence[{4, -4, 1}, {1, 2, 4}, 40] (* Vincenzo Librandi, Sep 30 2017 *)
|
|
PROG
|
(Sage) [lucas_number1(n, 3, 1)+1 for n in range(40)] # Zerinvary Lajos, Jul 06 2008
(PARI) vector(40, n, n--; fibonacci(2*n)+1) \\ G. C. Greubel, Jun 06 2019
(GAP) List([0..40], n-> Fibonacci(2*n)+1 ) # G. C. Greubel, Jun 06 2019
|
|
CROSSREFS
|
Apart from the first term, same as A052925.
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|