A010099 -id:A010099 - cp's OEIS frontend

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

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.

A010100 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=10.

Original entry on oeis.org

1, 10, 10, 100, 1000, 100000, 100000000, 10000000000000, 1000000000000000000000, 10000000000000000000000000000000000, 10000000000000000000000000000000000000000000000000000000

Offset: 0

N. J. A. Sloane

nonn

From Peter Bala, Nov 11 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 := 9*sum {n = 1..inf} 1/10^floor(n*phi) (= 81*sum {n = 1..inf} floor(n/phi)/10^n) = 0.90990 90990 99090 99090 ... = 1/(1 + 1/(10 + 1/(10 + 1/(100 + 1/(1000 + 1/(100000 + 1/(100000000 + ...))))))). The constant c is known to be transcendental (see Adams and Davison 1977). Cf. A014565 and A005614.
Furthermore, for k = 0,1,2,... if we define the real number X(k) = sum {n >= 1} 1/10^(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)

Alois P. Heinz, 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. A005614, A014565, A214706, A214887, A215270, A215271, A215272.

Column k=10 of A244003.

a:= n-> 10^(<<1|1>, <1|0>>^n)[1, 2]:
seq(a(n), n=0..12);  # Alois P. Heinz, Jun 17 2014

10^Fibonacci[Range[0,10]] (* Harvey P. Dale, Feb 12 2023 *)

a(n) = 10^fibonacci(n); \\ Michel Marcus, Oct 25 2017

a(n) = 10^Fibonacci(n).

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).

A076776 a(0) = 1, a(1) = 2, a(2) = 5; for n > 2, a(n) = a(n-1)*a(n-2).

Original entry on oeis.org

1, 2, 5, 10, 50, 500, 25000, 12500000, 312500000000, 3906250000000000000, 1220703125000000000000000000000, 4768371582031250000000000000000000000000000000000

Offset: 0

Emily Shields (emilyshields_2001(AT)hotmail.com), Nov 14 2002

nonn

Cf. A000045, A000301, A010098, A010099.

with(combinat, fibonacci):A076776 := n->2^fibonacci(n-2)*5^fibonacci(n-1);

nxt[{a_,b_}]:={b,a*b}; Join[{1},NestList[nxt,{2,5},15][[All,1]]] (* Harvey P. Dale, Jun 07 2021 *)

a(n) = 2^fibonacci(n-2)*5^fibonacci(n-1)for n>=2, fibonacci(n)=A000045(n). - Vladeta Jovovic and Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 16 2002

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Nov 16 2002

cp's OEIS Frontend

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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A010100 a(n) = a(n-1)*a(n-2) with a(0)=1, a(1)=10.

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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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