A010100 -id:A010100 - cp's OEIS frontend

A005614 The binary complement of the infinite Fibonacci word A003849. Start with 1, apply 0->1, 1->10, iterate, take limit.

1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 0

Offset: 0

N. J. A. Sloane

Previous name was: The infinite Fibonacci word (start with 1, apply 0->1, 1->10, iterate, take limit).
Characteristic function of A022342. - Philippe Deléham, May 03 2004
a(n) = number of 0's between successive 1's (see also A003589 and A007538). - Eric Angelini, Jul 06 2005
With offset 1 this is the characteristic sequence for Wythoff A-numbers A000201=[1,3,4,6,...].
Eric Angelini's comment made me think that if 1 is defined to be the number of 0's between successive 1's in a string of 0's and 1's, then this string is 101. Applying the same operation to the digits of 101 leads to 101101, the iteration leads to successive palindromes of lengths given by A001911, up to a(n). - Rémi Schulz, Jul 06 2010
For generalized Fibonacci words see A221150, A221151, A221152, ... - Peter Bala, Nov 11 2013
The limiting mean of the first n terms is phi - 1; the limiting variance is phi (A001622). - Clark Kimberling, Mar 12 2014
Apply the difference operator to every column of the Wythoff difference array, A080164, to get an array of Fibonacci numbers, F(h). Replace each F(h) with h, and apply the difference operator to every column. In the resulting array, every column is A005614. - Clark Kimberling, Mar 02 2015
Binary expansion of the rabbit constant A014565. - M. F. Hasler, Nov 10 2018

			The infinite word is 101101011011010110101101101011...

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003.
G. Melançon, Factorizing infinite words using Maple, MapleTech journal, vol. 4, no. 1, 1997, pp. 34-42, esp. p. 36.

T. D. Noe, Table of n, a(n) for n = 0..10945 (19 iterations)
Ricardo Gómez Aíza, Symbolic dynamical scales: modes, orbitals, and transversals, arXiv:2009.02669 [math.DS], 2020.
F. Axel et al., Vibrational modes in a one dimensional "quasi-alloy": the Morse case, J. de Physique, Colloq. C3, Supp. to No. 7, Vol. 47 (Jul 1986), pp. C3-181-C3-186; see Eq. (10).
Joerg Arndt, Matters Computational (The Fxtbook), 753-754.
E. A. Bender and J. T. Butler, Asymptotic approximations for the number of fanout-free functions, IEEE Trans. Computers, 27 (1978), 1180-1183. (Annotated scanned copy)
M. Bunder and K. Tognetti, On the self matching properties of [j tau], Discrete Math., 241 (2001), 139-151.
J. T. Butler, Letter to N. J. A. Sloane, Dec. 1978.
Glen Joyce C. Dulatre, Jamilah V. Alarcon, Vhenedict M. Florida, and Daisy Ann A. Disu, On Fractal Sequences, DMMMSU-CAS Science Monitor (2016-2017) Vol. 15 No. 2, 109-113.
S. Dulucq and D. Gouyou-Beauchamps, Sur les facteurs des suites de Sturm, Theoret. Comput. Sci. 71 (1990), 381-400.
M. S. El Naschie, Statistical geometry of a Cantor discretum and semiconductors, Computers & Mathematics with Applications, Vol. 29 (Issue 12, June 1995), 103-110.
D. Gault and M. Clint, "Curiouser and curiouser" said Alice. Further reflections on an interesting recursive function, Internat. J. Computer Math., 26 (1988), 35-43.
D. Gault and M. Clint, "Curiouser and curiouser said Alice. Further reflections on an interesting recursive function, Intern. J. Computer. Math., 26 (1988), 35-43. (Annotated scanned copy)
J. Grytczuk, Infinite semi-similar words, Discrete Math. 161 (1996), 133-141.
Clark Kimberling, Intriguing infinite words composed of zeros and ones, Elemente der Mathematik (2021).
Clark Kimberling and K. B. Stolarsky, Slow Beatty sequences, devious convergence, and partitional divergence, Amer. Math. Monthly, 123 (No. 2, 2016), 267-273.
K. L. Kodandapani and S. C. Seth, On combinational networks with restricted fan-out, IEEE Trans. Computers, 27 (1978), 309-318. (Annotated scanned copy)
Ron Knott, The Fibonacci Rabbit Sequence
Wolfdieter Lang, The Tribonacci and ABC Representations of Numbers are Equivalent, arXiv preprint arXiv:1810.09787 [math.NT], 2018.
G. Melançon, Lyndon factorization of sturmian words, Discr. Math., 210 (2000), 137-149.
S. Mneimneh, Fibonacci in The Curriculum: Not Just a Bad Recurrence, in Proceeding SIGCSE '15 Proceedings of the 46th ACM Technical Symposium on Computer Science Education, 253-258.
C. Mongoven, The Rabbit Sequence (a musical composition with A005614).
T. D. Noe, The first 1652 subwords, including leading zeros.
José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake, arXiv preprint arXiv:1212.1368 [cs.DM], 2012-2014.
Jeffrey Shallit, Characteristic words as fixed points of homomorphisms, University of Waterloo Technical Report CS-91-72, 1991.
Jeffrey Shallit, Characteristic words as fixed points of homomorphisms. [Cached copy, with permission]
N. J. A. Sloane, Families of Essentially Identical Sequences, Mar 24 2021 (Includes this sequence)
K. B. Stolarsky, Beatty sequences, continued fractions and certain shift operators, Canad. Math. Bull., 19 (1976), 473-482.
Scott V. Tezlaf, On ordinal dynamics and the multiplicity of transfinite cardinality, arXiv:1806.00331 [math.NT], 2018. See p. 10.
Eric Weisstein's World of Mathematics, Rabbit Constant and Rabbit Sequence.
Index entries for characteristic functions
Index entries for sequences that are fixed points of mappings

Binary complement of A003849, which is the standard form of this sequence.
Two other essentially identical sequences are A096270, A114986.
Subwords: A178992, A171676.
Cf. A000045 (Fibonacci numbers), A001468, A001911, A005206 (partial sums), A014565, A014675, A022342, A036299, A044432, A221150, A221151, A221152.
Cf. A339051 (odd bisection), A339052 (even bisection).
The following sequences are all essentially the same, in the sense that they are simple transformations of each other, with A000201 as the parent: A000201, A001030, A001468, A001950, A003622, A003842, A003849, A004641, A005614, A014675, A022342, A088462, A096270, A114986, A124841. - N. J. A. Sloane, Mar 11 2021

a005614 n = a005614_list !! n
a005614_list = map (1 -) a003849_list
-- Reinhard Zumkeller, Apr 07 2012

[Floor((n+1)*(-1+Sqrt(5))/2)-Floor(n*(-1+Sqrt(5))/2): n in [1..100]]; // Vincenzo Librandi, Jan 17 2019

Digits := 50; u := evalf((1-sqrt(5))/2); A005614 := n->floor((n+1)*u)-floor(n*u);

Nest[ Flatten[ # /. {0 -> {1}, 1 -> {1, 0}}] &, {1}, 10] (* Robert G. Wilson v, Jan 30 2005 *)
Flatten[Nest[{#, #[[1]]} &, {1, 0}, 9]] (* IWABUCHI Yu(u)ki, Oct 23 2013 *)
SubstitutionSystem[{0 -> {1}, 1 -> {1, 0}}, {1, 0}, 9] // Last (* Jean-François Alcover, Feb 06 2020 *)

a(n,w1,s0,s1)=local(w2); for(i=2,n,w2=[ ]; for(k=1,length(w1),w2=concat(w2, if(w1[ k ],s1,s0))); w1=w2); w2
for(n=2,10,print(n" "a(n,[ 0 ],[ 1 ],[ 1,0 ]))) \\ Gives successive convergents to sequence

/* for m>=1 compute exactly A183136(m+1)+1 terms of the sequence */
r=(1+sqrt(5))/2;v=[1,0];for(n=2,m,v=concat(v,vector(floor((n+1)/r),i,v[i]));a(n)=v[n];) /* Benoit Cloitre, Jan 16 2013 */

from math import isqrt
def A005614(n): return (n+isqrt(m:=5*(n+2)**2)>>1)-(n+1+isqrt(m-10*n-15)>>1) # Chai Wah Wu, Aug 17 2022

Define strings S(0)=1, S(1)=10, thereafter S(n)=S(n-1)S(n-2); iterate. Sequence is S(oo). The individual S(n)'s are given in A036299.
a(n) = floor((n+2)*u) - floor((n+1)*u), where u = (-1 + sqrt(5))/2.
Sum_{n>=0} a(n)/2^(n+1) = A014565. - R. J. Mathar, Jul 19 2013
From Peter Bala, Nov 11 2013: (Start)
If we read the present sequence as the digits of a decimal constant c = 0.101101011011010 ... then we have the series representation c = Sum_{n >= 1} 1/10^floor(n*phi). An alternative representation is c = Sum_{n >= 1} 1/10^floor(n/phi) - 10/9.
The constant 9*c has the simple continued fraction representation [0; 1, 10, 10, 100, 1000, ..., 10^Fibonacci(n), ...]. See A010100.
Using this result we can find the alternating series representation c = 1/9 - 9*Sum_{n >= 1} (-1)^(n+1)*(1 + 10^Fibonacci(3*n+1))/( (10^(Fibonacci(3*n - 1)) - 1)*(10^(Fibonacci(3*n + 2)) - 1) ). The series converges very rapidly: for example, the first 10 terms of the series give a value for c accurate to more than 5.7 million decimal places. Cf. A014565. (End)
a(n) = A005206(n+1) - A005206(n). a(2*n) = A339052(n); a(2*n+1) = A339051(n+1). - Peter Bala, Aug 09 2022

Corrected by Clark Kimberling, Oct 04 2000

Name corrected by Michel Dekking, Apr 02 2019

A000301 a(n) = a(n-1)*a(n-2) with a(0) = 1, a(1) = 2; also a(n) = 2^Fibonacci(n).

Original entry on oeis.org

1, 2, 2, 4, 8, 32, 256, 8192, 2097152, 17179869184, 36028797018963968, 618970019642690137449562112, 22300745198530623141535718272648361505980416, 13803492693581127574869511724554050904902217944340773110325048447598592

Offset: 0

N. J. A. Sloane, Mar 15 1996

Continued fraction expansion of s = A073115 = 1.709803442861291... = Sum_{k >= 0} (1/2^floor(k * phi)) where phi is the golden ratio (1 + sqrt(5))/2. - Benoit Cloitre, Aug 19 2002

The continued fraction expansion of the above constant s is [1; 1, 2, 2, 4, ...], that of the rabbit constant r = s-1 = A014565 is [0; 1, 2, 2, 4, ...]. - M. F. Hasler, Nov 10 2018

Stephen Wolfram, A New Kind of Science, Wolfram Media, 2002, p. 913.

T. D. Noe, Table of n, a(n) for n = 0..18
Manosij Ghosh Dastidar and Michael Wallner, Bijections between Variants of Dyck Paths and Integer Compositions, arXiv:2406.16404 [math.CO], 2024. See p. 1.
J. L. Davison, A series and its associated continued fraction, Proc. Amer. Math. Soc., 63 (1977), 29-32.
Samuele Giraudo, Intervals of balanced binary trees in the Tamari lattice, arXiv preprint arXiv:1107.3472 [math.CO], 2011-2012, and Theor Comput Sci 420 (2012) 1-27.
Bertrand Teguia Tabuguia, Computing with D-Algebraic Sequences, arXiv:2412.20630 [math.AG], 2024. See p. 9.
Index to divisibility sequences

Cf. A000045, A000079, A000304, A010098, A010099, A010100, A073115, A124091.

Column k = 2 of A244003.

a000301 = a000079 . a000045
a000301_list = 1 : scanl (*) 2 a000301_list
-- Reinhard Zumkeller, Mar 20 2013

[2^Fibonacci(n): n in [0..20]]; // Vincenzo Librandi, Apr 18 2011

A000301 := proc(n) option remember;
             if n < 2 then 1+n
           else A000301(n-1)*A000301(n-2)
             fi
           end:
seq(A000301(n), n=0..15);

2^Fibonacci[Range[0, 14]] (* Alonso del Arte, Jul 28 2016 *)

a(n)=1<Charles R Greathouse IV, Jan 12 2012

[2^fibonacci(n) for n in range(15)] # G. C. Greubel, Jul 29 2024

a(n) ~ k^phi^n with k = 2^(1/sqrt(5)) = 1.3634044... and phi the golden ratio. - Charles R Greathouse IV, Jan 12 2012
a(n) = A000304(n+3) / A010098(n+1). - Reinhard Zumkeller, Jul 06 2014
Sum_{n>=0} 1/a(n) = A124091. - Amiram Eldar, Oct 27 2020
Limit_{n->oo} a(n)/a(n-1)^phi = 1. - Peter Woodward, Nov 24 2023

Offset changed from 1 to 0 by Vincenzo Librandi, Apr 18 2011

A010098 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=3.

Original entry on oeis.org

1, 3, 3, 9, 27, 243, 6561, 1594323, 10460353203, 16677181699666569, 174449211009120179071170507, 2909321189362570808630465826492242446680483, 507528786056415600719754159741696356908742250191663887263627442114881

Offset: 0

N. J. A. Sloane

nonn

From Peter Bala, Nov 01 2013: (Start)
Let phi = (1/2)*(1 + sqrt(5)) denote the golden ratio A001622. This sequence gives the simple continued fraction expansion of the constant c := 2*Sum_{n>=1} 1/3^floor(n*phi) (= 4*Sum_{n>=1} floor(n/phi)/3^n) = 0.768597560593155198508 ... = 1/(1 + 1/(3 + 1/(3 + 1/(9 + 1/(27 + 1/(243 + 1/(6561 + ...))))))). The constant c is known to be transcendental (see Adams and Davison 1977). Cf. A014565.
Furthermore, for k = 0,1,2,... if we put X(k) = sum {n >= 1} 1/3^(n*Fibonacci(k) + Fibonacci(k+1)*floor(n*phi)) then the real number X(k+1)/X(k) has the simple continued fraction expansion [0; a(k+1), a(k+2), a(k+3), ...] (apply Bowman 1988, Corollary 1). (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..17
W. W. Adams and J. L. Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), 194-198.
P. G. Anderson, T. C. Brown, and P. J.-S. Shiue, A simple proof of a remarkable continued fraction identity, Proc. Amer. Math. Soc. 123 (1995), 2005-2009.
D. Bowman, A new generalization of Davison's theorem, Fib. Quart. Volume 26 (1988), 40-45

Cf. A000045, A000301, A000304, A010099, A010100, A014565, A214706, A214887, A215270, A215271, A215272.

Column k=3 of A244003.

a010098 n = a010098_list !! n
a010098_list = 1 : 3 : zipWith (*) a010098_list (tail a010098_list)
-- Reinhard Zumkeller, Jul 06 2014

[3^Fibonacci(n): n in [0..12]]; // G. C. Greubel, Jul 29 2024

a[-1]:=1: a[0]:=3: a[1]:=3: for n from 2 to 13 do a[n]:=a[n-1]*a[n-2] od: seq(a[n], n=-1..10); # Zerinvary Lajos, Mar 19 2009

3^Fibonacci[Range[0,13]] (* Vladimir Joseph Stephan Orlovsky, Jan 21 2012 *)
RecurrenceTable[{a[0]==1,a[1]==3,a[n]==a[n-1]a[n-2]},a,{n,15}] (* Harvey P. Dale, Jan 21 2021 *)

[3^fibonacci(n) for n in range(13)] # G. C. Greubel, Jul 29 2024

a(n) = 3^Fibonacci(n).

a(n+1) = A000304(n+3) / A000301(n). - Reinhard Zumkeller, Jul 06 2014

A244003 A(n,k) = k^Fibonacci(n); square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 2, 1, 0, 1, 4, 3, 4, 1, 0, 1, 5, 4, 9, 8, 1, 0, 1, 6, 5, 16, 27, 32, 1, 0, 1, 7, 6, 25, 64, 243, 256, 1, 0, 1, 8, 7, 36, 125, 1024, 6561, 8192, 1, 0, 1, 9, 8, 49, 216, 3125, 65536, 1594323, 2097152, 1, 0

Offset: 0

Alois P. Heinz, Jun 17 2014

			Square array A(n,k) begins:
  1, 1,   1,    1,     1,      1,       1, ...
  0, 1,   2,    3,     4,      5,       6, ...
  0, 1,   2,    3,     4,      5,       6, ...
  0, 1,   4,    9,    16,     25,      36, ...
  0, 1,   8,   27,    64,    125,     216, ...
  0, 1,  32,  243,  1024,   3125,    7776, ...
  0, 1, 256, 6561, 65536, 390625, 1679616, ...

Alois P. Heinz, Antidiagonals n = 0..20, flattened

Columns k=0-10 give: A000007, A000012, A000301, A010098, A010099, A214706, A215270, A214887, A215271, A215272, A010100.
Rows n=0, 1+2, 3-8 give: A000012, A001477, A000290, A000578, A000584, A001016, A010801, A010809.
Main diagonal gives: A152915.
Cf. A000045, A103323.

A:= (n, k)-> k^(<<1|1>, <1|0>>^n)[1, 2]:
seq(seq(A(n, d-n), n=0..d), d=0..12);

A[0, 0] = 1; A[n_, k_] := k^Fibonacci[n]; Table[A[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Nov 11 2015 *)

A(n,k) = k^A000045(n).

A(0,k) = 1, A(1,k) = k, A(n,k) = A(n-1,k) * A(n-2,k) for n>=2.

A214706 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=5.

Original entry on oeis.org

1, 5, 5, 25, 125, 3125, 390625, 1220703125, 476837158203125, 582076609134674072265625, 277555756156289135105907917022705078125, 161558713389263217748322010169914619837072677910327911376953125

Offset: 0

Vincenzo Librandi, Aug 01 2012

a(17) has 1117 digits.
From Peter Bala, Nov 01 2013: (Start)
Let phi = 1/2*(1 + sqrt(5)) denote the golden ratio A001622. This sequence is the simple continued fraction expansion of the constant c := 4*Sum_{n = 1..oo} 1/5^floor(n*phi) (= 16*Sum_{n = 1..oo} floor(n/phi)/5^n) = 0.83866 83869 91037 14262 ... = 1/(1 + 1/(5 + 1/(5 + 1/(25 + 1/(125 + 1/(3125 + 1/(390625 + ...))))))). The constant c is known to be transcendental (see Adams and Davison 1977). Cf. A014565.
Furthermore, for k = 0,1,2,... if we define the real number X(k) = sum {n >= 1} 1/5^(n*Fibonacci(k) + Fibonacci(k+1)*floor(n*phi)) then the real number X(k+1)/X(k) has the simple continued fraction expansion [0; a(k+1), a(k+2), a(k+3), ...] (apply Bowman 1988, Corollary 1). (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..16
W. W. Adams and J. L. Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), 194-198.
P. G. Anderson, T. C. Brown, and P. J.-S. Shiue, A simple proof of a remarkable continued fraction identity, Proc. Amer. Math. Soc. 123 (1995), 2005-2009.
D. Bowman, A new generalization of Davison's theorem, Fib. Quart. Volume 26 (1988), 40-45.

Cf. A000045, A000301, A001622, A010098, A010099, A010100, A014565, A214706, A214887, A215270, A215271, A215272.

Column k=5 of A244003.

Magma
```
[5^Fibonacci(n): n in [0..13]];
```

a:= n-> 5^(<<1|1>, <1|0>>^n)[1, 2]:
seq(a(n), n=0..12);  # Alois P. Heinz, Jun 17 2014

5^Fibonacci[Range[0,11]]
nxt[{a_,b_}]:={b,a*b}; NestList[nxt,{1,5},12][[All,1]] (* Harvey P. Dale, Oct 14 2018 *)

[5^fibonacci(n) for n in range(15)] # G. C. Greubel, Jan 07 2024

a(n) = 5^Fibonacci(n).

A214887 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=7.

Original entry on oeis.org

1, 7, 7, 49, 343, 16807, 5764801, 96889010407, 558545864083284007, 54116956037952111668959660849, 30226801971775055948247051683954096612865741943

Offset: 0

Vincenzo Librandi, Aug 01 2012

a(17) has 1350 digits.
From Peter Bala, Nov 01 2013: (Start)
Let phi = 1/2*(1 + sqrt(5)) denote the golden ratio A001622. This sequence is the simple continued fraction expansion of the constant c := 6*sum {n = 1..inf} 1/7^floor(n*phi) (= 36*sum {n = 1..inf} floor(n/phi)/7^n) = 0.87718 67194 00499 51922 ... = 1/(1 + 1/(7 + 1/(7 + 1/(49 + 1/(343 + 1/(16807 + 1/(5764801 + ...))))))). The constant c is known to be transcendental (see Adams and Davison 1977). Cf. A014565.
Furthermore, for k = 0,1,2,... if we define the real number X(k) = sum {n >= 1} 1/7^(n*Fibonacci(k) + Fibonacci(k+1)*floor(n*phi)) then the real number X(k+1)/X(k) has the simple continued fraction expansion [0; a(k+1), a(k+2), a(k+3), ...] (apply Bowman 1988, Corollary 1). (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..16
W. W. Adams and J. L. Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), 194-198.
P. G. Anderson, T. C. Brown, P. J.-S. Shiue, A simple proof of a remarkable continued fraction identity, Proc. Amer. Math. Soc. 123 (1995), 2005-2009.
D. Bowman, A new generalization of Davison's theorem, Fib. Quart. Volume 26 (1988), 40-45

Cf. A000045, A000301, A010098, A010099, A010100, A214706, A014565, A215270, A215271, A215272.

Column k=7 of A244003.

Magma
```
[7^Fibonacci(n): n in [0..10]];
```

a:= n-> 7^(<<1|1>, <1|0>>^n)[1, 2]:
seq(a(n), n=0..12);  # Alois P. Heinz, Jun 17 2014

7^Fibonacci[Range[0,10]]
nxt[{a_,b_}]:={b,a*b}; Transpose[NestList[nxt,{1,7},10]][[1]] (* Harvey P. Dale, Jun 10 2014 *)

a(n) = 7^Fibonacci(n).

A215270 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=6.

Original entry on oeis.org

1, 6, 6, 36, 216, 7776, 1679616, 13060694016, 21936950640377856, 286511799958070431838109696, 6285195213566005335561053533150026217291776, 1800782593726645086383198950649858141454002621435149880441896326660096

Offset: 0

Bruno Berselli, Aug 07 2012

From Peter Bala, Nov 01 2013: (Start)
Let phi = 1/2*(1 + sqrt(5)) denote the golden ratio A001622. This sequence is the simple continued fraction expansion of the constant c := 5*sum {n = 1..inf} 1/6^floor(n*phi) (= 25*sum {n = 1..inf} floor(n/phi)/6^n) = 0.86045 01626 86090 61353 ... = 1/(1 + 1/(6 + 1/(6 + 1/(36 + 1/(216 + 1/(7776 + 1/(1679616 + ...))))))). The constant c is known to be transcendental (see Adams and Davison 1977). Cf. A014565.
Furthermore, for k = 0,1,2,... if we define the real number X(k) = sum {n >= 1} 1/6^(n*Fibonacci(k) + Fibonacci(k+1)*floor(n*phi)) then the real number X(k+1)/X(k) has the simple continued fraction expansion [0; a(k+1), a(k+2), a(k+3), ...] (apply Bowman 1988, Corollary 1). (End)

Bruno Berselli, Table of n, a(n) for n = 0..15
W. W. Adams and J. L. Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), 194-198.
P. G. Anderson, T. C. Brown, P. J.-S. Shiue, A simple proof of a remarkable continued fraction identity, Proc. Amer. Math. Soc. 123 (1995), 2005-2009.
D. Bowman, A new generalization of Davison's theorem, Fib. Quart. Volume 26 (1988), 40-45

Cf. A000045, A000301, A010098-A010100, A214706, A214887, A215271, A215272.
Cf. A166470 (same recurrence with initial values 2, 6). A014565.
Column k=6 of A244003.

Magma
```
[6^Fibonacci(n): n in [0..11]];
```

a:= n-> 6^(<<1|1>, <1|0>>^n)[1, 2]:
seq(a(n), n=0..12);  # Alois P. Heinz, Jun 17 2014

RecurrenceTable[{a[0] == 1, a[1] == 6, a[n] == a[n - 1] a[n - 2]}, a[n], {n, 0, 15}]

a(n) = 6^Fibonacci(n).

A215271 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=8.

Original entry on oeis.org

1, 8, 8, 64, 512, 32768, 16777216, 549755813888, 9223372036854775808, 5070602400912917605986812821504, 46768052394588893382517914646921056628989841375232, 237142198758023568227473377297792835283496928595231875152809132048206089502588928

Offset: 0

Bruno Berselli, Aug 07 2012

From Peter Bala, Nov 01 2013: (Start)
Let phi = 1/2*(1 + sqrt(5)) denote the golden ratio A001622. This sequence is the simple continued fraction expansion of the constant c := 7*sum {n = 1..inf} 1/8^floor(n*phi) (= 49*sum {n = 1..inf} floor(n/phi)/8^n) = 0.89040 80325 60827 28336 ... = 1/(1 + 1/(8 + 1/(8 + 1/(64 + 1/(512 + 1/(32768 + 1/(16777216 + ...))))))). The constant c is known to be transcendental (see Adams and Davison 1977). Cf. A014565.
Furthermore, for k = 0,1,2,... if we define the real number X(k) = sum {n >= 1} 1/8^(n*Fibonacci(k) + Fibonacci(k+1)*floor(n*phi)) then the real number X(k+1)/X(k) has the simple continued fraction expansion [0; a(k+1), a(k+2), a(k+3), ...] (apply Bowman 1988, Corollary 1). (End)

Bruno Berselli, Table of n, a(n) for n = 0..15
W. W. Adams and J. L. Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), 194-198.
P. G. Anderson, T. C. Brown, P. J.-S. Shiue, A simple proof of a remarkable continued fraction identity, Proc. Amer. Math. Soc. 123 (1995), 2005-2009.
D. Bowman, A new generalization of Davison's theorem, Fib. Quart. Volume 26 (1988), 40-45

Cf. A000045, A000301, A010098-A010100, A214706, A214887, A215270, A215272, A014565.

Column k=8 of A244003.

Magma
```
[8^Fibonacci(n): n in [0..11]];
```

a:= n-> 8^(<<1|1>, <1|0>>^n)[1, 2]:
seq(a(n), n=0..12);  # Alois P. Heinz, Jun 17 2014

RecurrenceTable[{a[0] == 1, a[1] == 8, a[n] == a[n - 1] a[n - 2]}, a[n], {n, 0, 15}]

a(n) = 8^Fibonacci(n).

A215272 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=9.

Original entry on oeis.org

1, 9, 9, 81, 729, 59049, 43046721, 2541865828329, 109418989131512359209, 278128389443693511257285776231761, 30432527221704537086371993251530170531786747066637049

Offset: 0

Bruno Berselli, Aug 07 2012

From Peter Bala, Nov 01 2013: (Start)
Let phi = 1/2*(1 + sqrt(5)) denote the golden ratio A001622. This sequence is the simple continued fraction expansion of the constant c := 8*sum {n = 1..inf} 1/9^floor(n*phi) (= 64*sum {n = 1..inf} floor(n/phi)/9^n) = 0.90109 74122 99938 29901 ... = 1/(1 + 1/(9 + 1/(9 + 1/(81 + 1/(729 + 1/(59049 + 1/(43046721 + ...))))))). The constant c is known to be transcendental (see Adams and Davison 1977). Cf. A014565.
Furthermore, for k = 0,1,2,... if we define the real number X(k) = sum {n >= 1} 1/9^(n*Fibonacci(k) + Fibonacci(k+1)*floor(n*phi)) then the real number X(k+1)/X(k) has the simple continued fraction expansion [0; a(k+1), a(k+2), a(k+3), ...] (apply Bowman 1988, Corollary 1). (End)

Bruno Berselli, Table of n, a(n) for n = 0..15
W. W. Adams and J. L. Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), 194-198.
P. G. Anderson, T. C. Brown, P. J.-S. Shiue, A simple proof of a remarkable continued fraction identity, Proc. Amer. Math. Soc. 123 (1995), 2005-2009.
D. Bowman, A new generalization of Davison's theorem, Fib. Quart. Volume 26 (1988), 40-45

Cf. A000045, A000301, A010098-A010100, A214706, A214887, A215270, A215271, A014565.

Column k=9 of A244003.

Magma
```
[9^Fibonacci(n): n in [0..10]];
```

a:= n-> 9^(<<1|1>, <1|0>>^n)[1, 2]:
seq(a(n), n=0..12);  # Alois P. Heinz, Jun 17 2014

RecurrenceTable[{a[0] == 1, a[1] == 9, a[n] == a[n - 1] a[n - 2]}, a[n], {n, 0, 15}]

a(n) = 9^fibonacci(n); \\ Jinyuan Wang, Apr 06 2019

a(n) = 9^Fibonacci(n).

A010099 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=4.

Original entry on oeis.org

1, 4, 4, 16, 64, 1024, 65536, 67108864, 4398046511104, 295147905179352825856, 1298074214633706907132624082305024, 383123885216472214589586756787577295904684780545900544

Offset: 0

N. J. A. Sloane

From Peter Bala, Nov 01 2013: (Start)
Let phi = 1/2*(1 + sqrt(5)) denote the golden ratio A001622. This sequence is the simple continued fraction expansion of the constant c := 3*sum {n = 1..inf} 1/4^floor(n*phi) (= 9*sum {n = 1..inf} floor(n/phi)/4^n) = 0.80938 42984 64421 90504 ... = 1/(1 + 1/(4 + 1/(4 + 1/(16 + 1/(64 + 1/(1024 + 1/(65536 + ...))))))). The constant c is known to be transcendental (see Adams and Davison 1977). Cf. A014565.
Furthermore, for k = 0,1,2,... if we define the real number X(k) = sum {n >= 1} 1/4^(n*Fibonacci(k) + Fibonacci(k+1)*floor(n*phi)) then the real number X(k+1)/X(k) has the simple continued fraction expansion [0; a(k+1), a(k+2), a(k+3), ...] (apply Bowman 1988, Corollary 1). (End)

Indranil Ghosh, Table of n, a(n) for n = 0..17
W. W. Adams and J. L. Davison, A remarkable class of continued fractions, Proc. Amer. Math. Soc. 65 (1977), 194-198.
P. G. Anderson, T. C. Brown, P. J.-S. Shiue, A simple proof of a remarkable continued fraction identity, Proc. Amer. Math. Soc. 123 (1995), 2005-2009.
D. Bowman, A new generalization of Davison's theorem, Fib. Quart. Volume 26 (1988), 40-45

Cf. A000045, A000301, A010098, A010100, A014565, A214706, A214887, A215270, A215271, A215272.

Column k=4 of A244003.

a[ -1]:=1:a[0]:=4: a[1]:=4: for n from 2 to 13 do a[n]:=a[n-1]*a[n-2] od: seq(a[n], n=-1..10); # Zerinvary Lajos, Mar 19 2009

a(n) = 4^Fibonacci(n).

cp's OEIS Frontend

A005614 The binary complement of the infinite Fibonacci word A003849. Start with 1, apply 0->1, 1->10, iterate, take limit.

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A000301 a(n) = a(n-1)*a(n-2) with a(0) = 1, a(1) = 2; also a(n) = 2^Fibonacci(n).

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A010098 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=3.

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A244003 A(n,k) = k^Fibonacci(n); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A214706 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=5.

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A214887 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=7.

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