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URL: https://oeis.org/A142975

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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.
2
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.
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*(x^4 - 2*x^3 + x^2 + 2*x - 1)/((x + 1)*(x^2 - 3*x + 1)*(x^2 + x - 1)).
E.g.f.: 1 + (exp(-x) - exp(x/2)*((5 + exp(x))*cosh(sqrt(5)*x/2) - sqrt(5)*(3 + exp(x))*sinh(sqrt(5)*x/2)))/5. - Stefano Spezia, Dec 26 2025
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
Sequence in context: A340766 A161943 A134184 * A211277 A114589 A192908
KEYWORD
nonn,easy
AUTHOR
Gregory Nisbet (gregory.nisbet(AT)gmail.com), Jul 15 2008
STATUS
approved