|
| |
|
|
A059929
|
|
Fib(n)*Fib(n+2).
|
|
10
| |
|
|
0, 2, 3, 10, 24, 65, 168, 442, 1155, 3026, 7920, 20737, 54288, 142130, 372099, 974170, 2550408, 6677057, 17480760, 45765226, 119814915, 313679522, 821223648, 2149991425, 5628750624, 14736260450, 38580030723
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,2
|
|
|
COMMENTS
| Expansion of golden ratio (1+sqrt(5))/2 as an infinite product: phi = prod(i=0, infty, (1+1/(fibonacci(2i+1) * fibonacci(2i+3)-1)) * (1-1/(fibonacci(2i+2) * fibonacci(2i+4)+1))) - Thomas Baruchel (baruchel(AT)users.sourceforge.net), Nov 11 2003
Each of these is one short of or one over the square of a Fibonacci number (A007598). This means that a rectangle sized F(n) by F(n + 2) units can't be converted into a square with sides of length F(n + 1) units unless one square unit of material is added or removed. - Alonso del Arte, May 03 2011
|
|
|
LINKS
| Harry J. Smith, Table of n, a(n) for n=0,...,500
M. Renault, Dissertation
M. Waldschmidt, Open Diophantine problems
E. H. Kuo, Applications of graphical condensation for enumerating matchings and tilings
YouTube, Engineering Bits & Bytes: The Fibonacci Puzzle
|
|
|
FORMULA
| a(n) = Fib(n+1)^2-(-1)^n = A007598(n+1)+A033999(n+1) = A000045(n+1)^2-A033999(n)
G.f.: [2x-x^2]/[(1+x)(1-3x+x^2)].
Sum[n=1..inf, 1/a(n)] = 1, Sum[n=1..inf, (-1)^n/a(n)] = 2-sqrt(5).
Sum[n=1..inf, 1/a(2n-1)] = 1/phi = (sqrt(5)-1)/2. - Franz Vrabec (franz.vrabec(AT)aon.at), Sep 15 2005
1 = 1/2 + 1/3 + 1/10 + 1/24 + 1/65 + 1/168 + ..., = 1/(1*2) + 1/(1*3) + 1/(2*5) + 1/(3*8) + ... - Gary W. Adamson (qntmpkt(AT)yahoo.com), Mar 16 2008
Sum[n=1..inf, 1/a(2n)] = (3-sqrt(5))/2. [From Franz Vrabec (franz.vrabec(AT)aon.at), Nov 30 2009]
a(n)=((7+3*sqrt(5))/10)*((3+sqrt(5))/2)^(n-1)+((7-3*sqrt(5))/10)*((3-sqrt(5))/2)^(n-1)+(3/5)*(-1)^(n-1). [Tim Monahan, Aug 09 2011]
|
|
|
MAPLE
| with (combinat):a:=n->fibonacci(n)*fibonacci(n+2): seq(a(n), n=0..26); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 07 2007
|
|
|
MATHEMATICA
| Table[Fibonacci[n]*Fibonacci[n+2], {n, 0, 60}] (* From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 17 2009 *)
|
|
|
PROG
| (PARI) { for (n=0, 500, write("b059929.txt", n, " ", fibonacci(n)*fibonacci(n + 2)); ) } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Jun 30 2009]
|
|
|
CROSSREFS
| Bisection of A070550.
First differences of A059840.
Sequence in context: A130002 A162034 A105286 * A123029 A103018 A005158
Adjacent sequences: A059926 A059927 A059928 * A059930 A059931 A059932
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| Henry Bottomley (se16(AT)btinternet.com), Feb 09 2001
|
| |
|
|
