A142975 n-th term of the Fibonacci-type sequence x(1)=1, x(2)=Fibonacci(n), x(k+1)=x(k)+x(k-1) for k>1.

1, 1, 3, 7, 17, 43, 109, 281, 727, 1891, 4929, 12871, 33641, 87985, 230203, 602447, 1576849, 4127635, 10805301, 28287049, 74053871, 193871371, 507555073, 1328785487, 3478787857, 9107556193, 23843845299, 62423922391, 163427829137, 427859414971, 1120150172989

Offset

1,3

Comments

Original definition:

This sequence is derived from the Fibonacci sequence. Think of it as the Fibonacci sequence applied to itself. The sequence is generated by taking the n-th element of a recursive sequence whose first terms are 1 and the n-th element Fibonacci term.

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (3,1,-5,-1,1).

Formula

a(n) = fib(1,fib(1,1,n),n).

a(n) = Fibonacci(n-1) * Fibonacci(n) + Fibonacci(n-2) = A001654(n-1)+A000045(n-2).

G.f.: (x^5-2*x^4+x^3+2*x^2-x) / (x^5-x^4-5*x^3+x^2+3*x-1).

Example

The fifth term, 17, would be computed by hand like this:
1) Generate the first five values of the Fibonacci sequence: 1,1,2,3,5
2) The result of step 1 is the second value of the Fibonacci-type sequence used in computing 17.
3) Thus we know the first two terms of our Fibonacci-type sequence: 1,5...
4) The sequence can be extended through recursive addition f(n) = f(n-1) + f(n- 2): 1,5,6,11,17
5) The fifth element of this sequence is 17 and thus the answer.

Maple

F:= combinat[fibonacci]:
seq(F(n-1)*F(n)+F(n-2), n=1..32);

Mathematica

fts[{a_, b_, c_}]:=b*c+a; fts/@Partition[Fibonacci[Range[-1, 30]], 3, 1] (* or *) LinearRecurrence[ {3, 1, -5, -1, 1}, {1, 1, 3, 7, 17}, 40] (* Harvey P. Dale, Nov 24 2016 *)

Prog

(Python)
def fib(arb1, arb2, nth):
    if nth == 0:
            return arb1
    if nth == 1:
            return arb2
    x = [0]*nth
    x[0] = arb1
    x[1] = arb2
    for i in range(2, nth, 1):
            x[i] = x[i-1]+x[i-2]
    return x[-1]
def fib2d(n):
    return fib(1, fib(1, 1, n), n)
[fib2d(i) for i in range(1, 10)]
# the function fib2d will return the n-th term of the sequence.

Crossrefs

Cf. A000045, A143077.

Sequence in context: A340766 A161943 A134184 * A211277 A114589 A192908

Adjacent sequences: A142972 A142973 A142974 * A142976 A142977 A142978

Keyword

nonn,easy

Author

Gregory Nisbet (gregory.nisbet(AT)gmail.com), Jul 15 2008

Status

approved