A058182 -id:A058182 - cp's OEIS frontend

A087417 Sum of the cubes of A058182.

-1, 0, 0, 1, 2, 10, 135, 19818, 395466722, 156401766357161409, 24461512581491800525933058683030176

Offset: 0

Miklos Kristof, Oct 22 2003

sign

The sequence A058182=b(n) has the property: b(1)^3+b(2)^3+b(3)^3+...b(n)^3=b(n)*b(n+1)

			A058182 begins 1,1,2,5,27,734,538783...=b(n)
1^3+1^3=2=a(2)=1*2=b(2)*b(3).
1^3+1^3+2^3=10=a(3)=2*5=b(3)*b(4).
1^3+1^3+2^3+5^3=135=a(4)=5*27=b(4)*b(5).
1^3+1^3+2^3+5^3+27^3=19818=a(5)=b(5)*b(6).

Cf. A058182.

{a(n) = local(a0, a1, a2); if( n<0, a(-n), if( n<3, -(n==0), a0 = a1 = 1; for(i=4, n, a2 = a1^2 + a0; a0 = a1; a1 = a2); a1*a0))}; /* Michael Somos, May 22 2005 */

a(n+1) = a(n) + A058182(n)^3, a(-n) = a(n) for all n in Z.

A000278 a(n) = a(n-1) + a(n-2)^2 for n >= 2 with a(0) = 0 and a(1) = 1.

Original entry on oeis.org

0, 1, 1, 2, 3, 7, 16, 65, 321, 4546, 107587, 20773703, 11595736272, 431558332068481, 134461531248108526465, 186242594112190847520182173826, 18079903385772308300945867582153787570051, 34686303861638264961101080464895364211215702792496667048327

Offset: 0

Stephen J. Greenfield (greenfie(AT)math.rutgers.edu)

nonn

T. D. Noe, Table of n, a(n) for n = 0..25
W. Duke, Stephen J. Greenfield, and Eugene R. Speer, Properties of a Quadratic Fibonacci Recurrence, J. Integer Seq. 1 (1998), Article #98.1.8.

Cf. A000283, A058182.

[n le 2 select n-1 else Self(n-1) + Self(n-2)^2: n in [1..18]]; // Vincenzo Librandi, Dec 17 2015

A000278 := proc(n) option remember; if n <= 1 then n else A000278(n-2)^2+A000278(n-1); fi; end;

Join[{a=0,b=1},Table[c=a^2+b;a=b;b=c,{n,16}]] (* Vladimir Joseph Stephan Orlovsky, Jan 22 2011 *)
RecurrenceTable[{a[n +2] == a[n +1] + a[n]^2, a[0] == 1, a[1] == 1}, a, {n, 0, 16}] (* Robert G. Wilson v, Apr 14 2017 *)

PARI
```
a(n)=if(n<2,n>0,a(n-1)+a(n-2)^2)
```

def A000278():
    x, y = 0, 1
    while True:
        yield x
        x, y = x + y, x * x
a = A000278(); [next(a) for i in range(18)]  # Peter Luschny, Dec 17 2015

a(2n) is asymptotic to A^(sqrt(2)^(2n-1)) where A=1.668751581493687393311628852632911281060730869124873165099170786836201970866312366402366761987... and a(2n+1) to B^(sqrt(2)^(2n)) where B=1.693060557587684004961387955790151505861127759176717820241560622552858106116817244440438308887... See reference for proof. - Benoit Cloitre, May 03 2003

Name edited by Petros Hadjicostas, Nov 03 2019

A001042 a(n) = a(n-1)^2 - a(n-2)^2.

Original entry on oeis.org

1, 2, 3, 5, 16, 231, 53105, 2820087664, 7952894429824835871, 63248529811938901240357985099443351745, 4000376523371723941902615329287219027543200136435757892789536976747706216384

Offset: 0

N. J. A. Sloane, R. K. Guy

The next term has 152 digits. - Franklin T. Adams-Watters, Jun 11 2009

Archimedeans Problems Drive, Eureka, 27 (1964), 6.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..13
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
R. K. Guy, Letters to N. J. A. Sloane, June-August 1968
R. P. Loh, A. G. Shannon, A. F. Horadam, Divisibility Criteria and Sequence Generators Associated with Fermat Coefficients, Preprint, 1980.
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A001146, A000278, A000283, A058182, A007018.

Cf. A064236 (numbers of digits).

a001042 n = a001042_list !! n
a001042_list = 1 : 2 : zipWith (-) (tail xs) xs
               where xs = map (^ 2) a001042_list
-- Reinhard Zumkeller, Dec 16 2013

RecurrenceTable[{a[0]==1,a[1]==2,a[n]==a[n-1]^2-a[n-2]^2},a,{n,0,12}] (* Harvey P. Dale, Jan 11 2013 *)

a(n) ~ c^(2^n), where c = 1.1853051643868354640833201434870139866230288004895868726506278977814490371... . - Vaclav Kotesovec, Dec 17 2014

More terms from James Sellers, Sep 19 2000.

A058181 Quadratic recurrence a(n) = a(n-1)^2 - a(n-2) for n >= 2 with a(0) = 1 and a(1) = 0.

Original entry on oeis.org

1, 0, -1, 1, 2, 3, 7, 46, 2109, 4447835, 19783236185116, 391376433956083065015485621, 153175513056180249189030531428945090978436751221570525

Offset: 0

Henry Bottomley, Nov 15 2000

sign

			a(6) = a(5)^2 - a(4) = 3^2 - 2 = 7.

Vincenzo Librandi, Table of n, a(n) for n = 0..16
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437.
A. V. Aho and N. J. A. Sloane, Some doubly exponential sequences, Fibonacci Quarterly, Vol. 11, No. 4 (1973), pp. 429-437 (original plus references that F.Q. forgot to include - see last page!)
Index entries for sequences of form a(n+1) = a(n)^2 + ...

Cf. A058182.

a:=[1,0];; for n in [3..15] do a[n]:=a[n-1]^2-a[n-2]; od; a; # G. C. Greubel, Jun 09 2019

I:=[1,0]; [n le 2 select I[n] else Self(n-1)^2 - Self(n-2): n in [1..15]]; // G. C. Greubel, Jun 09 2019

Join[{a=1,b=0},Table[c=b^2-a;a=b;b=c,{n,13}]] (* Vladimir Joseph Stephan Orlovsky, Jan 18 2011 *)
RecurrenceTable[{a[0]==1, a[1]==0, a[n]==a[n-1]^2 - a[n-2]}, a, {n, 13}] (* Vincenzo Librandi, Nov 11 2012 *)

a(n)=if(n<0, a(-1-n), if(n<2, 1-n, a(n-1)^2-a(n-2))) /* Michael Somos, May 05 2005 */

def a(n):
    if (n==0): return 1
    elif (n==1): return 0
    else: return a(n-1)^2 - a(n-2)
[a(n) for n in (0..15)] # G. C. Greubel, Jun 09 2019

a(n)^2 = a(n+1) + a(n-1), a(-1-n) = a(n).
For n >= 4, a(n) = ceiling(c^(2^n)) with c=1.0303497388742578142745024606710866\
16436302563960998408889321488508667424048981473368773165340730475719244472111...
and c^(1/4) = 1.0075025785879710605024343257517358... - Benoit Cloitre, Apr 16 2007

A000284 a(n) = a(n-1)^3 + a(n-2) with a(0)=0, a(1)=1.

Original entry on oeis.org

0, 1, 1, 2, 9, 731, 390617900, 59601394712394173339000731, 211723599072542785377729319366442939995427829921816290889198752331804918235791

Offset: 0

Stephen J. Greenfield (greenfie(AT)math.rutgers.edu)

John Cerkan, Table of n, a(n) for n = 0..10

Cf. A058182.

A000284 := proc(n) option remember; if n <= 1 then n else A000284(n-2)+A000284(n-1)^3; fi; end;
a[-2]:=0: a[-1]:=1: a[0]:=1: a[1]:=2: for n from 2 to 6 do a[n]:=a[n-1]^3+a[n-2] od: seq(a[n], n=-2..6); # Zerinvary Lajos, Mar 19 2009

RecurrenceTable[{a[n] == a[n-1]^3 + a[n-2], a[0] == 0, a[1] == 1}, a, {n, 0, 8}] (* Jean-François Alcover, Feb 06 2016 *)
nxt[{a_,b_}]:={b,b^3+a}; NestList[nxt,{0,1},9][[All,1]] (* Harvey P. Dale, May 08 2020 *)

For n>0, a(n) = floor(c^(3^n)) where c=1.0275090796393628012075291021962112731026759143420911102331653107087209649910... - Gerald McGarvey, Nov 28 2007

a(9) from Vincenzo Librandi, Apr 26 2010

A292433 a(0) = 0, a(1) = 1; a(n) = prime(a(n-1))*a(n-1) + a(n-2).

Original entry on oeis.org

0, 1, 2, 7, 121, 79988, 81600798165, 182421074243967704954243

Offset: 0

Ilya Gutkovskiy, Dec 08 2017

			+---+-------------+--------------------+-------------------+
| n | a(n)/a(n+1) | Continued fraction |      Comment      |
+---+-------------+--------------------+-------------------+
| 1 |    1/2      | [0; 2]             |   2 = prime(a(1)) |
+---+-------------+--------------------+-------------------+
| 2 |    2/7      | [0; 3, 2]          |   3 = prime(a(2)) |
+---+-------------+--------------------+-------------------+
| 3 |    7/121    | [0; 17, 3, 2]      |  17 = prime(a(3)) |
+---+-------------+--------------------+-------------------+
| 4 |  121/79988  | [0; 661, 17, 3, 2] | 661 = prime(a(4)) |
+---+-------------+--------------------+-------------------+

Cf. A000040, A000278, A006277, A007097, A036247, A036248, A058182, A076146, A083659.

RecurrenceTable[{a[0] == 0, a[1] == 1, a[n] == Prime[a[n - 1]] a[n - 1] + a[n - 2]}, a[n], {n, 7}]

A307799 a(0) = 0, a(1) = 3; a(n) = rev(a(n-1))*a(n-1) + a(n-2), where rev = digit reversal (A004086).

Original entry on oeis.org

0, 3, 9, 84, 4041, 5673648, 48020423368761, 806086788756824484462571572, 221815145293562950532110825781341443907408910699844537

Offset: 0

Ilya Gutkovskiy, Apr 29 2019

The next term is too large to include.

			+---+--------------+---------------------+------------------+
| n | a(n)/a(n+1)  | Continued fraction  |      Comment     |
+---+--------------+---------------------+------------------+
| 1 |    3/9       | [0; 3]              |    3 = rev(a(1)) |
+---+--------------+---------------------+------------------+
| 2 |    9/84      | [0; 9, 3]           |    9 = rev(a(2)) |
+---+--------------+---------------------+------------------+
| 3 |   84/4041    | [0; 48, 9, 3]       |   48 = rev(a(3)) |
+---+--------------+---------------------+------------------+
| 4 | 4041/5673648 | [0; 1404, 48, 9, 3] | 1404 = rev(a(4)) |
+---+--------------+---------------------+------------------+

Cf. A001040, A004086, A058182.

a[n_] := a[n] = FromDigits[Reverse[IntegerDigits[a[n - 1]]]] a[n - 1] + a[n - 2]; a[0] = 0; a[1] = 3; Table[a[n], {n, 0, 8}]

cp's OEIS Frontend

A087417 Sum of the cubes of A058182.

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A000278 a(n) = a(n-1) + a(n-2)^2 for n >= 2 with a(0) = 0 and a(1) = 1.

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A001042 a(n) = a(n-1)^2 - a(n-2)^2.

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A058181 Quadratic recurrence a(n) = a(n-1)^2 - a(n-2) for n >= 2 with a(0) = 1 and a(1) = 0.

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A000284 a(n) = a(n-1)^3 + a(n-2) with a(0)=0, a(1)=1.

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A292433 a(0) = 0, a(1) = 1; a(n) = prime(a(n-1))*a(n-1) + a(n-2).

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A307799 a(0) = 0, a(1) = 3; a(n) = rev(a(n-1))*a(n-1) + a(n-2), where rev = digit reversal (A004086).

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