A005371 -id:A005371 - cp's OEIS frontend

A081255 Duplicate of A005371.

3, 1, 4, 7, 29, 199, 5778, 1149851, 6643838879, 7639424778862807

Offset: 0

dead

A007570 a(n) = F(F(n)), where F is a Fibonacci number.

0, 1, 1, 1, 2, 5, 21, 233, 10946, 5702887, 139583862445, 1779979416004714189, 555565404224292694404015791808, 2211236406303914545699412969744873993387956988653, 2746979206949941983182302875628764119171817307595766156998135811615145905740557

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

a(20) is approximately 2.830748520089124 * 10^1413, much too large to include even in the b-file. - Alonso del Arte, Apr 30 2020
Let M(0) denote the 2 X 2 identity matrix, and let M(1) = [[0, 1], [1, 1]]. Let M(n) = M(n-2) * M(n-1). Then a(n) is equal to both the (1, 2)-entry and the (2, 1)-entry of M(n). - John M. Campbell, Jul 02 2021
This is a strong divisibility sequence, that is, gcd(a(n),a(m)) = a(gcd(n,m)) for n, m >= 1. - Peter Bala, Dec 06 2022

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..19 (terms n = 0..17 from T. D. Noe)
Alonso del Arte, Table of n, a(n) for n = 0 .. 24, with digits grouped in hundreds
John M. Campbell, A Matrix-Based Recursion Relation for F_{F_n}, Fib. Quart., Vol. 60, No. 3 (2022), pp. 256-261.
Bakir Farhi, Summation of certain infinite Fibonacci related series, arXiv:1512.09033 [math.NT], 2015. See p. 6, eq. 2.9.
George Ledin, Jr., Problem H-147, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 6, No. 6 (1968), p. 352; Converging Fractions, Solution to Problem H-147 by David Zeitlin, ibid., Vol. 8, No. 4 (1970), pp. 387-389.
Edward A. Parberry, Two recursion relations for F(F(n)), Fib. Quart., Vol. 15, No. 2 (1977), p. 122 and p. 139.
Martin Stein, Algebraic independence results for reciprocal sums of Fibonacci and Lucas numbers, Dissertation, Hannover: Gottfried Wilhelm Leibniz Universität Hannover, 2012.
Chris Street, A Recurrence for the Sequence {F(F(n)), n>=0}.

Cf. A000045, A001622, A005371, A058051, A094214.

F:= n-> (<<0|1>, <1|1>>^n)[1, 2]:
a:= n-> F(F(n)):
seq(a(n), n=0..14);  # Alois P. Heinz, Oct 09 2015

F[0] = 0; F[1] = 1; F[n_] := F[n] = F[n - 1] + F[n - 2]; Table[F[F[n]], {n, 0, 14}]
Fibonacci[Fibonacci[Range[0, 20]]] (* Harvey P. Dale, May 05 2012 *)

a(n)=fibonacci(fibonacci(n)) \\ Charles R Greathouse IV, Feb 03 2014

from sympy import fibonacci
def a(n): return fibonacci(fibonacci(n))
print([a(n) for n in range(15)]) # Michael S. Branicky, Feb 02 2022

[fibonacci(fibonacci(n)) for n in range(0, 14)] # Zerinvary Lajos, Nov 30 2009

val fibo: LazyList[BigInt] = (0: BigInt) #:: (1: BigInt) #:: fibo.zip(fibo.tail).map { n => n._1 + n._2 }
val fiboLimited: LazyList[Int] = 0 #:: 1 #:: fiboLimited.zip(fiboLimited.tail).map { n => n._1 + n._2 } // Limited to 32-bit integers because that's the type for LazyList apply()
(0 to 19).map(n => fibo(fiboLimited(n))) // Alonso del Arte, Apr 30 2020

a(n+1)/a(n) ~ phi^(F(n-1)), with phi = (1 + sqrt(5))/2 = A001622. - Carmine Suriano, Jan 24 2011
Sum_{n>=1} 1/a(n) = 3.7520024260... is transcendental (Stein, 2012). - Amiram Eldar, Oct 30 2020
Sum_{n>=1} (-1)^(F(n)+1)*a(n-1)/(a(n)*a(n+1)) = 1/phi (A094214) (Farhi, 2015). - Amiram Eldar, Apr 07 2021
Limit_{n->oo} a(n+1)/a(n)^phi = 5^((phi-1)/2) = 1.6443475285..., where phi is the golden ratio (A001622) (Ledin, 1968) - Amiram Eldar, Feb 02 2022

One more term from Harvey P. Dale, May 05 2012

A338736 a(n) = L(L(n)) mod n, where L = Lucas numbers = A000032.

Original entry on oeis.org

0, 0, 1, 1, 4, 0, 3, 7, 7, 4, 10, 3, 9, 10, 7, 15, 12, 0, 10, 9, 7, 4, 22, 3, 1, 4, 7, 1, 4, 18, 30, 31, 7, 4, 29, 15, 1, 34, 34, 39, 35, 24, 29, 29, 7, 4, 46, 3, 1, 4, 7, 29, 29, 0, 21, 55, 7, 54, 35, 3, 45, 4, 7, 63, 64, 36, 2, 29, 7, 4, 6, 3, 43, 4, 7, 29

Offset: 1

Alois P. Heinz, Nov 05 2020

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000032, A005371, A263112, A338638, A338889.

b:= proc(n) local r, M, p; r, M, p:=
      <<1|0>, <0|1>>, <<0|1>, <1|1>>, n;
      do if irem(p, 2, 'p')=1 then r:=
        `if`(nargs=1, r.M, r.M mod args[2]) fi;
         if p=0 then break fi; M:=
        `if`(nargs=1, M.M, M.M mod args[2])
      od; (r.<<2, 1>>)[1$2]
    end:
a:= n-> (f-> b(f, n) mod n)(b(n)):
seq(a(n), n=1..80);

a(n) = A005371(n) mod n.

A338889 a(n) = L(L(L(n))) mod L(L(n)), where L = Lucas numbers = A000032.

Original entry on oeis.org

1, 0, 3, 1, 1, 1, 0, 1, 1, 29, 7, 1, 19679776435706023589554718882448088434898811874077010905231927243854, 1, 7

Offset: 0

Alois P. Heinz, Nov 14 2020

nonn

a(21) = 2992285359..7163788371 has 5090 decimal digits.

Alois P. Heinz, Table of n, a(n) for n = 0..20

Cf. A000032, A005371, A262361, A274996, A338638, A338736.

b:= proc(n) local r, M, p; r, M, p:=
      <<1|0>, <0|1>>, <<0|1>, <1|1>>, n;
      do if irem(p, 2, 'p')=1 then r:=
        `if`(nargs=1, r.M, r.M mod args[2]) fi;
         if p=0 then break fi; M:=
        `if`(nargs=1, M.M, M.M mod args[2])
      od; (r.<<2, 1>>)[1$2]
    end:
a:= n-> (h-> b(h$2) mod h)(b(b(n))):
seq(a(n), n=0..15);

a(n) = A262361(n) mod A005371(n).

A068096 a(n) = F(L(n)) where F(n) = n-th Fibonacci number and L(n) = n-th Lucas number.

Original entry on oeis.org

1, 1, 2, 3, 13, 89, 2584, 514229, 2971215073, 3416454622906707, 22698374052006863956975682, 173402521172797813159685037284371942044301, 8801063578447437644962364569698707634360652047981718378070013667111

Offset: 0

Leroy Quet, Mar 22 2002

Vincenzo Librandi, Table of n, a(n) for n = 0..17

Cf. A000032, A000045, A005371.

[Fibonacci(Lucas(n)): n in [0..12]]; // Vincenzo Librandi, Sep 18 2017

Table[Fibonacci[LucasL[n]], {n, 0, 13}] (* Vladimir Joseph Stephan Orlovsky, Feb 08 2012 *)

a(n) = fibonacci(fibonacci(n+1)+fibonacci(n-1)) \\ Felix Fröhlich, Sep 17 2017

a(n) = Sum_{k=1..F(n-1)+1} binomial(F(n-1), k-1)*F(F(n)+k-1), where F(n) is A000045. - Tony Foster III, Sep 03 2018

A262361 a(n) = L(L(L(n))), where L(n) are Lucas numbers A000032.

Original entry on oeis.org

4, 1, 7, 29, 1149851, 387739824812222466915538827541705412334749

Offset: 0

Alois P. Heinz, Nov 09 2016

nonn

a(6) = 3393011755..4322744978 has 1208 decimal digits and a(7) = 4437405101..8830136999 has 240305 decimal digits.

Alois P. Heinz, Table of n, a(n) for n = 0..5

Cf. A000032, A005371, A058051.

L:= n-> (<<0|1>, <1|1>>^n. <<2, 1>>)[1$2]:
a:= n-> L(L(L(n))):
seq(a(n), n=0..5);

A262361 = Nest[LucasL, #, 3] &; Array[A262361, 6, 0] (* JungHwan Min, Nov 09 2016 *)

from sympy import lucas as L
def a(n):  return L(L(L(n)))
print([a(n) for n in range(6)]) # Michael S. Branicky, Apr 01 2021

A338638 a(n) = L(L(n)) mod L(n), where L = Lucas numbers = A000032.

Original entry on oeis.org

1, 0, 1, 3, 1, 1, 0, 1, 1, 7, 4, 1, 199, 1, 4, 843, 1, 1, 0, 1, 29, 123, 4, 1, 3, 199, 4, 39603, 29, 1, 5778, 1, 1, 7, 4, 17622890, 12752043, 1, 4, 39603, 7881196, 1, 5778, 1, 29, 7, 4, 1, 3, 1149851, 28143689044, 7, 29, 1, 0, 312119004790, 6643838879, 7, 4, 1

Offset: 0

Alois P. Heinz, Nov 04 2020

Alois P. Heinz, Table of n, a(n) for n = 0..4795

Cf. A000032, A005371, A016089, A213060, A263101, A338736, A338889.

b:= proc(n) local r, M, p; r, M, p:=
      <<1|0>, <0|1>>, <<0|1>, <1|1>>, n;
      do if irem(p, 2, 'p')=1 then r:=
        `if`(nargs=1, r.M, r.M mod args[2]) fi;
         if p=0 then break fi; M:=
        `if`(nargs=1, M.M, M.M mod args[2])
      od; (r.<<2, 1>>)[1$2]
    end:
a:= n-> (f-> b(f$2) mod f)(b(n)):
seq(a(n), n=0..60);

Table[Mod[LucasL[LucasL[n]],LucasL[n]],{n,0,60}] (* Harvey P. Dale, Jul 04 2022 *)

a(n) = A005371(n) mod A000032(n).

a(n) = 0 for n in { A016089 }.

A005372 a(n) = L(L(n+1)+1), where L(n) are Lucas numbers A000032.

Original entry on oeis.org

3, 7, 11, 47, 322, 9349, 1860498, 10749957122, 12360848946698171, 82123488809519507169850807, 627376215338105766356982006981782561278127, 31842547163971605907183271059340725709462269514762215168643703957079

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..16

Cf. A000032, A005371.

[Lucas(1+Lucas(n+1)): n in [0..15]]; // G. C. Greubel, Nov 14 2022

L:= n-> (<<0|1>, <1|1>>^(n). <<2,1>>)[1,1]:
a:= n-> L(L(n+1)+1):
seq(a(n), n=0..12);  # Alois P. Heinz, Jun 01 2016

LucasL[1 +LucasL[Range[16]]] (* G. C. Greubel, Nov 14 2022 *)

[lucas_number2(1+lucas_number2(n+1, 1,-1),1,-1) for n in range(15)] # G. C. Greubel, Nov 14 2022

a(8) onwards corrected by Sean A. Irvine, Jun 01 2016

Name edited by Alois P. Heinz, Jun 01 2016

cp's OEIS Frontend

A081255 Duplicate of A005371.

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A007570 a(n) = F(F(n)), where F is a Fibonacci number.

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A338736 a(n) = L(L(n)) mod n, where L = Lucas numbers = A000032.

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A338889 a(n) = L(L(L(n))) mod L(L(n)), where L = Lucas numbers = A000032.

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A068096 a(n) = F(L(n)) where F(n) = n-th Fibonacci number and L(n) = n-th Lucas number.

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Magma

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A262361 a(n) = L(L(L(n))), where L(n) are Lucas numbers A000032.

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Python

A338638 a(n) = L(L(n)) mod L(n), where L = Lucas numbers = A000032.

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Programs

Maple

Mathematica

Formula

A005372 a(n) = L(L(n+1)+1), where L(n) are Lucas numbers A000032.

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