A000301 -id:A000301 - cp's OEIS frontend

A299399 a(n) = a(n-1)a(n-2)a(n-3)*a(n-4); a(0..3) = (1, 1, 2, 3).

1, 1, 2, 3, 6, 36, 1296, 839808, 235092492288, 9211413321697223245824, 2356948205087252000835395074931259831484416, 4286423488783965214900384842824017360544199884413056912194095171350270745233063936

Offset: 0

M. F. Hasler, Apr 22 2018

nonn

A variant of A000336 which uses initial values (1,2,3,4).

A multiplicative variant of the tetranacci sequences A000078, A001631 and other variants.

Seiichi Manyama, Table of n, a(n) for n = 0..14

Cf. A000336 (variant starting 1,2,3,4).
Cf. A000301 (order 2 variant), A000308 (order 3 variant).
Subsequence of A003586 (3-smooth numbers).
Cf. A000078, A001631 (additive variants).

nxt[{a_,b_,c_,d_}]:={b,c,d,a b c d}; NestList[nxt,{1,1,2,3},13][[All,1]] (* Harvey P. Dale, Jun 09 2022 *)

A299399(n,a=[1,1,2,3,6])={for(n=5,n,a[n%#a+1]=a[(n-1)%#a+1]^2\a[n%#a+1]);a[n%#a+1]}

a(n) = a(n-1)^2 / a(n-5) for n > 4.

a(n) = 2^A001631(n)*3^A000078(n).

A036544 a(n) = (2*(1 + n + (((10^n-1)/9) - n)/9)).

Original entry on oeis.org

2, 4, 8, 32, 256, 2480, 24704, 246928, 2469152, 24691376, 246913600, 2469135824, 24691358048, 246913580272, 2469135802496, 24691358024720, 246913580246944

Offset: 0

Antti Karttunen

Result of applying extrapolator to terms 3-7 of A000301.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
A. Karttunen, Lisp Source (see function g and sequence cddr_of_mult2last)
Index entries for linear recurrences with constant coefficients, signature (12, -21, 10).

[(2*(1+n+(((10^n-1)/9)-n)/9)): n in [0..30]]; // Vincenzo Librandi, Nov 11 2011

LinearRecurrence[{12,-21,10},{2,4,8},20] (* Harvey P. Dale, Mar 11 2025 *)

a(n) = (2/81)*(72*n + 10^n + 80).

G.f.: 2*(1 - 10*x + x^2)/((1-10*x)*(1-x)^2). - Bruno Berselli, Nov 11 2011

A073116 Continued fraction expansion of S/2 where S = Sum_{k>=0} 1/2^floor(k*phi) (A073115) and phi is the golden ratio (1+sqrt(5))/2 (A001622).

Original entry on oeis.org

0, 1, 5, 1, 8, 4, 64, 128, 16384, 1048576, 34359738368, 18014398509481984, 1237940039285380274899124224, 11150372599265311570767859136324180752990208, 27606985387162255149739023449108101809804435888681546220650096895197184

Offset: 1

Benoit Cloitre, Aug 19 2002

The number S is the number whose digits are obtained from the substitution system (1->(1,0),0->(1)). The n-th term of the continued fraction expansion for S is 2^Fibonacci(n-2) (cf. A000301). This number S is known to be transcendental. The continued fraction of S/2^m follows the same kind of rule as for S/2.

Amiram Eldar, Table of n, a(n) for n = 1..20

Cf. A000045, A000301, A001622, A073115.

a[1] = 0; a[2] = 1; a[3] = 5; a[n_] := 2^(Fibonacci[n - 2] - (-1)^n); Array[a, 15] (* Amiram Eldar, May 08 2022 *)

If n>2, a(2n+1) = 2^(F(2n-1)+1) and a(2n)= 2^(F(2n-2)-1), where F(n) is the n-th Fibonacci number.

More terms from Amiram Eldar, May 08 2022

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

A168089 a(n) = 2^pentanacci(n).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 4, 16, 256, 65536, 2147483648, 2305843009213693952, 1329227995784915872903807060280344576, 110427941548649020598956093796432407239217743554726184882600387580788736

Offset: 0

Vladimir Joseph Stephan Orlovsky, Nov 18 2009

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..16

Cf. A001591, A000301, A063401, A168088.

2^LinearRecurrence[{1, 1, 1, 1, 1}, {0, 0, 0, 0, 1}, 15] (* Vincenzo Librandi, Dec 09 2018 *)

a(n) = 2^A001591(n).

A249055 a(1)=0; the next term is always the product of the two smallest numbers not yet in the sequence and which have not yet been used.

Original entry on oeis.org

0, 2, 12, 30, 56, 90, 143, 210, 272, 342, 420, 506, 600, 702, 812, 992, 1122, 1260, 1406, 1560, 1722, 1892, 2070, 2256, 2450, 2652, 2862, 3135, 3422, 3660, 3906, 4160, 4422, 4692, 4970, 5256, 5550, 5852, 6162, 6480, 6806, 7140, 7482, 7832, 8372, 8742, 9120, 9506, 9900

Offset: 1

N. J. A. Sloane, Nov 01 2014

nonn

Suggested by A075336 and A249406.

			Start with a(1) = 0. The missing numbers are 1 2 3 4 5 6 ...
Multiply the first two, and we get 2, which is therefore a(2).
Cross 1, 2, and 1*2 = 2 off the missing list.
The first two missing numbers are now 3 and 4, so a(3) = 3*4 = 12.
Cross off 3,4,12 from the missing list.
Repeat!

Cf. A075326, A000045, A249406, A000304, A000301.

M:=50; A:=[0]; miss:=[seq(n,n=1..M^2)]:
for n from 1 to M do t1:=miss[1]*miss[2]; A:=[op(A),t1];
miss:=[seq(miss[i],i=3..nops(miss))];
miss:=remove('x->x=t1',miss);
od:
A;

Typo in definition corrected by Douglas Latimer, Nov 01 2014

A258161 a(n) = a(n-1)^3/a(n-2) with a(0)=1, a(1)=2.

Original entry on oeis.org

1, 2, 8, 256, 2097152, 36028797018963968, 22300745198530623141535718272648361505980416

Offset: 0

Vaclav Kotesovec, May 22 2015

The next term has 114 digits.

Cf. A000045, A000301, A001906, A064098, A208207, A230900, A258169.

Clear[a]; RecurrenceTable[{a[n]==a[n-1]^3/a[n-2], a[0]==1, a[1]==2},a,{n,0,8}]
Clear[a]; a[0]=2; a[n_]:=a[n]=Product[a[j]^(n-j),{j,0,n-1}]; Flatten[{1, Table[a[n], {n,1,8}]}]
Table[2^(Fibonacci[2*n]), {n, 0, 8}]

a(n) = 2^(Fibonacci(2*n)).

A228778 a(n) = 2^Fibonacci(n) + 1.

Original entry on oeis.org

2, 3, 3, 5, 9, 33, 257, 8193, 2097153, 17179869185, 36028797018963969, 618970019642690137449562113, 22300745198530623141535718272648361505980417, 13803492693581127574869511724554050904902217944340773110325048447598593

Offset: 0

Yeshwant Shivrai Valaulikar and M. Tamba, Sep 04 2013

nonn

Cf. A000045, A000301, A063896.

a:= n-> 1 + 2^(<<0|1>, <1|1>>^n)[1,2]:
seq(a(n), n=0..15);  # Alois P. Heinz, Aug 12 2017

Table[2^Fibonacci[n] + 1, {n, 0, 13}] (* T. D. Noe, Sep 07 2013 *)

a(n+2) = a(n+1)*a(n) - a(n) - a(n+1) + 2, a(0)=2, a(1)=3.
Binet type formula: log_2(a(n)-1) = (1/sqrt(5)) * (r^n - s^n), where r and s are the roots of x^2-x-1. (this is true by definition).
a(n) = A000301(n) + 1 = A063896(n) + 2. - Alois P. Heinz, Aug 12 2017

A277121 Numbers that immediately halt Conway's FIBONACCIGAME.

Original entry on oeis.org

1, 2, 4, 5, 7, 8, 10, 11, 14, 16, 20, 22, 25, 28, 32, 35, 40, 44, 47, 49, 50, 53, 55, 56, 59, 61, 64, 67, 70, 71, 73, 77, 79, 80, 83, 88, 89, 94, 97, 98, 100, 101, 103, 106, 107, 109, 110, 112, 113, 118, 121, 122, 125, 127, 128, 131, 134, 137, 139, 140, 142, 146, 149, 151, 154, 157, 158

Offset: 1

Alonso del Arte, Sep 30 2016

nonn

Like PRIMEGAME, Conway's FIBONACCIGAME uses a list of fractions to come up with a sequence expressed as exponents of powers of 2, the sequence in this case being the Fibonacci numbers (A000045, see A000301).
Unlike PRIMEGAME, FIBONACCIGAME does not have any integers in its list of rational numbers, which means that the process always comes to a halt, sooner or later, depending on the initial value.
Hence this sequence includes all powers of 2 (A000079). Also all primes greater than 43.
However, this sequence includes no multiples of 13 (as they can be handled by 17/65 or 1/13), nor any multiples of 3 (as they're taken care of by 1/3).
Indeed the sequence contains no multiples of 17, 19, 23, 29, 31, 37 either.

Cf. A275483/A275484.

is(n)=gcd(16990599132039,n)==1 && n%65 && n%34 && n%69 && n%341 && n%287 \\ Charles R Greathouse IV, Nov 26 2016

A377231 a(n) = digital root of 2^Fibonacci(n).

Original entry on oeis.org

1, 2, 2, 4, 8, 5, 4, 2, 8, 7, 2, 5, 1, 5, 5, 7, 8, 2, 7, 5, 8, 4, 5, 2, 1, 2, 2, 4, 8, 5, 4, 2, 8, 7, 2, 5, 1, 5, 5, 7, 8, 2, 7, 5, 8, 4, 5, 2, 1, 2, 2, 4, 8, 5, 4, 2, 8, 7, 2, 5, 1, 5, 5, 7, 8, 2, 7, 5, 8, 4, 5, 2, 1, 2, 2, 4, 8, 5, 4, 2, 8, 7, 2, 5, 1, 5, 5, 7, 8, 2, 7, 5, 8, 4, 5, 2

Offset: 0

Robert Bruce Gray, Oct 20 2024

This sequence has period 24.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A000301, A010888.

Module[{a, n}, RecurrenceTable[{a[n] == Mod[a[n-1]*a[n-2] - 1, 9] + 1, a[0] == 1, a[1] == 2}, a, {n, 0, 99}]] (* or *)
PadRight[{}, 100, {1, 2, 2, 4, 8, 5, 4, 2, 8, 7, 2, 5, 1, 5, 5, 7, 8, 2, 7, 5, 8, 4, 5, 2}] (* Paolo Xausa, Nov 08 2024 *)

a(n) = (a(n-1) * a(n-2) - 1) mod 9 + 1.

a(n) = A010888(A000301(n)).

cp's OEIS Frontend

A299399 a(n) = a(n-1)*a(n-2)*a(n-3)*a(n-4); a(0..3) = (1, 1, 2, 3).

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A036544 a(n) = (2*(1 + n + (((10^n-1)/9) - n)/9)).

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A073116 Continued fraction expansion of S/2 where S = Sum_{k>=0} 1/2^floor(k*phi) (A073115) and phi is the golden ratio (1+sqrt(5))/2 (A001622).

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

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A168089 a(n) = 2^pentanacci(n).

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A249055 a(1)=0; the next term is always the product of the two smallest numbers not yet in the sequence and which have not yet been used.

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A258161 a(n) = a(n-1)^3/a(n-2) with a(0)=1, a(1)=2.

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A228778 a(n) = 2^Fibonacci(n) + 1.

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A299399 a(n) = a(n-1)a(n-2)a(n-3)*a(n-4); a(0..3) = (1, 1, 2, 3).