A177727 a(0)=1; a(n) = a(n-1) * Fibonacci(3+n) * Fibonacci(1+n) / (Fibonacci(n))^2, n > 1.

1, 3, 30, 180, 1300, 8736, 60333, 412335, 2829310, 19384200, 132882696, 910735488, 6242420665, 42785803515, 293259265950, 2010026277756, 13776931957468, 94428478367520, 647222466507045, 4436128656563175, 30405678471399166, 208403619747957648, 1428419662108160400

Offset: 0

Author

Roger L. Bagula, May 12 2010

Keywords

• nonn
• easy

Comments

Similar recurrences a(n) = a(n-1)*F(a0+n-1)*F(b0+n-1)/(F(n)*F(c0+n-1)) are:

{a0,b0,c0} = {3,2,1} in A066258.

{a0,b0,c0} = {3,1,1} in A001654.

{a0,b0,c0} = {4,1,1} in A001655 and next for 5,6 as well.

References

• Harry Hochstadt, The Functions of Mathematical Physics, Dover, New York, 1986, p. 93.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Index entries for linear recurrences with constant coefficients, signature (5,15,-15,-5,1)

Crossrefs

Cf. A066258, A001654, A001655, A001656, A001657.

Programs

Magma

I:=[1, 3, 30, 180, 1300]; [n le 5 select I[n] else 5*Self(n-1)+15*Self(n-2)-15*Self(n-3)-5*Self(n-4)+Self(n-5): n in [1..30]]; // Vincenzo Librandi, Nov 18 2011

Maple

with (combinat):
A177727 := proc(n)
   if n = 0 then
           1;
   else
           procname(n-1)*fibonacci(3+n)*fibonacci(1+n)/fibonacci(n)^2 ;
   end if;
end proc:
seq(A177727(n),n=0..10) ; # R. J. Mathar, Nov 17 2011

Mathematica

a0 = 4; b0 = 2; c0 = 1;
a[0] = 1;
a[n_] := a[n] = (Fibonacci[(a0 + n - 1)]*Fibonacci[( b0 + n - 1)]/(Fibonacci[n]*Fibonacci[(c0 + n - 1)]))*a[n - 1];
Table[a[n], {n, 0, 30}]
LinearRecurrence[{5,15,-15,-5,1},{1,3,30,180,1300},30] (* Vincenzo Librandi, Nov 18 2011 *)

Formula

G.f.: ( -1+2*x ) / ( (x-1)*(x^2+3*x+1)*(x^2-7*x+1) ). - R. J. Mathar, Nov 17 2011

a(n) = A001656(n) - 2*A001656(n-1). - R. J. Mathar, Nov 17 2011