A000301 -id:A000301 - cp's OEIS frontend

A063401 a(n) = a(n-1)a(n-2)a(n-3) with a(0)=1, a(1)=2, a(2)=2.

1, 2, 2, 4, 16, 128, 8192, 16777216, 17592186044416, 2417851639229258349412352, 713623846352979940529142984724747568191373312, 30354201441027016733116592294117482916287606860189680019559568902170379456331382784

Offset: 0

Henry Bottomley, Jul 16 2001

nonn

			a(6) = 128*16*4 = 8192.

Harry J. Smith, Table of n, a(n) for n=0,...,15

Cf. A000301, A000308.

a0=1;a1=1;a2=2;lst={a0,a1,a2};Do[AppendTo[lst,a=a0*a1*a2]; a0=a1;a1=a2;a2=a, {n,12}];lst (* Vladimir Joseph Stephan Orlovsky, Nov 18 2009 *)
RecurrenceTable[{a[0]==1,a[1]==a[2]==2,a[n]==a[n-1]a[n-2]a[n-3]},a,{n,12}] (* Harvey P. Dale, Sep 05 2021 *)

{ for (n = 0, 15, if (n>2, a=a1*a2*a3; a3=a2; a2=a1; a1=a, if (n==0, a=a3=1; a1=a2=2, a=2)); write("b063401.txt", n, " ", a) ) } \\ Harry J. Smith, Aug 20 2009

a(n) = 2^A000073(n+1).

Definition corrected to a(1)=2 by Harry J. Smith, Aug 20 2009

A174677 a(n) = 2a(n-1)a(n-2) with a(0)=1 and a(1)=1.

Original entry on oeis.org

1, 1, 2, 4, 16, 128, 4096, 1048576, 8589934592, 18014398509481984, 309485009821345068724781056, 11150372599265311570767859136324180752990208

Offset: 0

Giovanni Teofilatto, Mar 26 2010

a(n) is the number of node minimal AVL trees of height n. - Alois P. Heinz, Mar 13 2013

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

Cf. A000045, A000301, A143897, A174666, A217298.

[2^(Fibonacci(n+1)-1): n in [0..10]]; // Vincenzo Librandi, Apr 24 2011

2^(Fibonacci[Range[15]] -1) (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)
Join[{1}, RecurrenceTable[{a[1]==1,a[2]==2,a[n]==2a[n-1]a[n-2]},a[n],{n, 12}]] (* Harvey P. Dale, Jul 07 2011 *)

[2^(fibonacci(n+1) -1) for n in range(13)] # G. C. Greubel, Jul 30 2024

a(n) = 2^(Fibonacci(n+1) - 1).

a(n) = 1/2 * A000301(n+1).

Formula index corrected by R. J. Mathar, Mar 30 2010

a(0)=1 prepended and name edited by Alois P. Heinz, Jul 05 2021

A011455 Sum 2^Fibonacci(i), i=2..n.

Original entry on oeis.org

2, 6, 14, 46, 302, 8494, 2105646, 17181974830, 36028814200938798, 618970019678718951650500910, 22300745198530623760505737951367313156481326, 13803492693581127574869511746854796103432841704846511061692361604079918

Offset: 2

Bagirath R. Krishnamachari (bagi(AT)miel.mot.com)

Ivan Panchenko, Table of n, a(n) for n = 2..18

Accumulate[2^Fibonacci[Range[2,15]]] (* Harvey P. Dale, May 07 2013 *)

Partial sums of A000301 (first 2 terms skipped). - R. J. Mathar, Apr 26 2007

Name corrected by T. D. Noe, Mar 03 2011

a(13) from Harvey P. Dale, May 07 2013

A061083 Fibonacci-type sequence based on division: a(0) = 1, a(1) = 2 and a(n) = a(n-2)/a(n-1) but ignore decimal point.

Original entry on oeis.org

1, 2, 5, 4, 125, 32, 390625, 8192, 476837158203125, 17179869184, 277555756156289135105907917022705078125, 618970019642690137449562112

Offset: 0

Ulrich Schimke (ulrschimke(AT)aol.com)

			a(6) = 390625, since a(4)/a(5) = 125/32 = 3.90625

Reinhard Zumkeller, Table of n, a(n) for n = 0..17

Cf. A061084 for subtraction, A000301 for multiplication and A000045 for addition - the common Fibonacci numbers

a061083 n = a061083_list !! n
a061083_list = 1 : 2 : zipWith divIgnPnt a061083_list (tail a061083_list)
   where divIgnPnt x y = ddiv (10 * m) x' where
            ddiv u w | r == 0    = 10 * w + q
                     | otherwise = ddiv (10 * r) (10 * w + q)
                     where (q,r) = divMod u y
            (x',m) = divMod x y
-- Reinhard Zumkeller, Dec 29 2011

a(n) = k^(n-th Fibonacci number) with k=2 if n is odd, k=5 if n is even

A089983 1, 1, 1, 1, ... a, b, c, d, ab-cd, ...

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 1, -1, 1, 2, -3, 5, 17, -91, 1532, 139497, -213710951, 29812036392235, 6371158648631364574889, -189937213493701003981668660072118562, 1210120120447335073097142485947209203511752911347585124133

Offset: 1

Ray Chandler, following a suggestion of Rainer Rosenthal, Nov 18 2003

Inspired by the formula for the determinant of a 2 X 2 matrix.

Sequence b(n,p) = a(n) (mod p), p prime, is a periodic sequence. Letting l(p) denotes the length of the period of b(n,p) we get l(2)=5, l(3)=11, l(5)=31... Is there any rule for l(p) ? - Benoit Cloitre, Nov 19 2003

Harvey P. Dale, Table of n, a(n) for n = 1..26

Cf. A089984.

I:=[1,1,1,1]; [n le 4 select I[n] else -Self(n-1)*Self(n-2)+Self(n-3)*Self(n-4): n in [1..22]]; // Vincenzo Librandi, Mar 30 2014

nxt[{a_,b_,c_,d_}]:={b,c,d,a b-c d}; NestList[nxt,{1,1,1,1},20][[All,1]] (* Harvey P. Dale, Oct 30 2021 *)

a=b=c=d=1;for(n=5,20,e=a*b-c*d;a=b;b=c;c=d;d=e;print1(e,","))

a(1)=a(2)=a(3)=a(4)=1, for n>4 a(n)=a(n-4)*a(n-3)-a(n-2)*a(n-1).

a(n) is asymptotic (in absolute value) to A^(phi^n) where phi=golden ratio and A=1.005384.. (follows same kind of behavior as A000301, A007660) - Benoit Cloitre, Nov 19 2003

A109213 Product of a(n-2) and digit reversal of a(n-1).

Original entry on oeis.org

1, 2, 2, 4, 8, 32, 184, 15392, 5400584, 74651892640, 25003708306137848, 6335942056759761366725617280, 20682199297864337408779128828731176793076928

Offset: 1

Zak Seidov, Jun 22 2005

Cf. A000301 a(n) = a(n-1)*a(n-2), A004086 R(n) = digit reversal of n, A109214 a(n)=a(n-1)*R(a(n-2)).

			a(5)=8, a(6)=32, R(32)=23, hence a(7)=8*23=184; R(184)=841, a(8)=32*841=15392, etc.

Cf. A000301, A004086, A109214.

a[1]=1;a[2]=2;a[n_]:=a[n]=a[n-2]*FromDigits[Reverse[IntegerDigits[a[n-1]]]]; A109213=Table[a[n], {n, 13}]
nxt[{a_,b_}]:={b,a*IntegerReverse[b]}; NestList[nxt,{1,2},15][[All,1]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 09 2017 *)

a(n)=a(n-2)*R(a(n-1)).

A109214 Product of a(n-1) and digit reversal of a(n-2).

Original entry on oeis.org

1, 2, 2, 4, 8, 32, 256, 5888, 3838976, 34109301760, 231888097227054080, 1556059601911449331359933440, 125186119679477750610733678211850458005934080, 55507466796083630515105997822341552764197877620395801846452095434158080

Offset: 1

Zak Seidov, Jun 22 2005

The next term (a(15)) has 115 digits. - Harvey P. Dale, Nov 14 2011

Cf. A000301 (a(n) = a(n-1)*a(n-2)), A004086 (R(n) = digit reversal of n), A109213 (a(n) = a(n-2)*R(a(n-1))).

R:= n-> (s-> parse(cat(s[-i]$i=1..length(s))))(""||n):
a:= proc(n) option remember; `if`(n<3, n, a(n-1)*R(a(n-2))) end:
seq(a(n), n=1..14);  # Alois P. Heinz, Sep 01 2025

a[1]=1;a[2]=2;a[n_]:=a[n]=a[n-1]*FromDigits[Reverse[IntegerDigits[a[n-2]]]]; A109214=Table[a[n], {n, 13}]
Transpose[NestList[{Last[#],Last[#]FromDigits[Reverse[ IntegerDigits[ First[ #]]]]}&,{1,2},13]][[1]] (* Harvey P. Dale, Nov 14 2011 *)

a(n) = a(n-1)*R(a(n-2)).

One more term (a(14)) from Harvey P. Dale, Nov 14 2011

A111235 a(1)=a(2)=a(3)=a(4)=1. For n >= 5, a(n)= a(n-1)a(n-2) + a(n-3)a(n-4).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 7, 23, 167, 3862, 645115, 2491437971, 1607264007306619, 4004398577225334507664179, 6436125704084114770053956998574742562466, 25772812612277833490303309040566300172816894832780792086674335463

Offset: 1

Leroy Quet, Oct 28 2005

a(5*n) is always even. Every other term of the sequence is odd.
It is easy to see that a(n) >= A000301(n-3) for all n. From that we can deduce that a(n) >= 2^(Fibonacci(n-3)). Can anybody give a formula for the asymptotic behavior? - Stefan Steinerberger, Jan 21 2006
As n->infinity, log(a(n))/phi^n approaches t-(-1)^n*u/phi^(2*n), where phi=(1+sqrt(5))/2, t=0.0672009781433377128..., and u=0.766475715574332057.... - Jon E. Schoenfield, Sep 14 2013

Alois P. Heinz, Table of n, a(n) for n = 1..21

Cf. A239967.

I:=[1,1,1,1]; [n le 4 select I[n] else Self(n-1)*Self(n-2) +Self(n-3)*Self(n-4): n in [1..16]]; // Vincenzo Librandi, Mar 30 2014

a:= proc(n) a(n):= `if`(n<5, 1, a(n-1)*a(n-2) +a(n-3)*a(n-4)) end:
seq(a(n), n=1..16);  # Alois P. Heinz, Mar 30 2014

RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==1,a[n]==a[n-1]a[n-2]+a[n-3] a[n-4]}, a,{n,20}] (* Harvey P. Dale, Jun 06 2017 *)

More terms from Stefan Steinerberger, Jan 21 2006

More terms from Joshua Zucker, May 04 2006

A275483 Numerators of Conway's FIBONACCIGAME.

Original entry on oeis.org

17, 133, 17, 23, 2233, 23, 31, 74, 31, 41, 129, 41, 13, 1, 1

Offset: 1

Alonso del Arte, Jul 30 2016

Like PRIMEGAME, the object of FIBONACCIGAME is to come up with a listing of the Fibonacci numbers through a process of multiplying integers by fractions to see which produce integers, which are then cycled back through the process.

However, unlike PRIMEGAME, FIBONACCIGAME starts from 78 * 5^(n - 1) and stops at 2^Fibonacci(n).

Julian Havil, Nonplussed! Mathematical Proof of Implausible Ideas. Princeton: Princeton University Press (2007): 174.

Cf. A275484 (denominators), A202138, A000301.

A275484 Denominators of Conway's FIBONACCIGAME.

Original entry on oeis.org

65, 34, 19, 17, 69, 29, 23, 341, 37, 31, 287, 43, 41, 13, 3

Offset: 1

Alonso del Arte, Jul 30 2016

Like PRIMEGAME, the object of FIBONACCIGAME is to come up with a listing of the Fibonacci numbers through a process of multiplying integers by fractions to see which produce integers, which are then cycled back through the process.

Notice that there isn't a 1 in this sequence, which means that the progress of the program, starting from a valid input, eventually halts with 2^Fibonacci(n).

Julian Havil, Nonplussed! Mathematical Proof of Implausible Ideas. Princeton: Princeton University Press (2007): 174.

Cf. A275483 (numerators), A203363, A000301.

cp's OEIS Frontend

A063401 a(n) = a(n-1)*a(n-2)*a(n-3) with a(0)=1, a(1)=2, a(2)=2.

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A174677 a(n) = 2*a(n-1)*a(n-2) with a(0)=1 and a(1)=1.

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A011455 Sum 2^Fibonacci(i), i=2..n.

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A061083 Fibonacci-type sequence based on division: a(0) = 1, a(1) = 2 and a(n) = a(n-2)/a(n-1) but ignore decimal point.

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A089983 1, 1, 1, 1, ... a, b, c, d, ab-cd, ...

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A109213 Product of a(n-2) and digit reversal of a(n-1).

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A109214 Product of a(n-1) and digit reversal of a(n-2).

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A111235 a(1)=a(2)=a(3)=a(4)=1. For n >= 5, a(n)= a(n-1)*a(n-2) + a(n-3)*a(n-4).

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A063401 a(n) = a(n-1)a(n-2)a(n-3) with a(0)=1, a(1)=2, a(2)=2.

A174677 a(n) = 2a(n-1)a(n-2) with a(0)=1 and a(1)=1.

A111235 a(1)=a(2)=a(3)=a(4)=1. For n >= 5, a(n)= a(n-1)a(n-2) + a(n-3)a(n-4).