A000283 -id:A000283 - cp's OEIS frontend

A014253 a(n) = b(n)^2, where b(n) = b(n-1)^2 + b(n-2)^2 (A000283).

0, 1, 1, 4, 25, 841, 749956, 563696135209, 317754178344723197077225, 100967717855888389973004528722798800700252204356

Offset: 0

nonn

G. C. Greubel, Table of n, a(n) for n = 0..13
Index entries for sequences of form a(n+1)=a(n)^2 + ...

[0] cat [n le 2 select 1 else (Self(n-1)+Self(n-2))^2: n in [1..10]]; // Vincenzo Librandi, Apr 02 2015

RecurrenceTable[{a[n]==(a[n-1]+a[n-2])^2, a[0]==0, a[1]==1}, a, {n,0,10}] (* G. C. Greubel, Jun 18 2019 *)

m=10; v=concat([0,1], vector(m-2)); for(n=3, m, v[n]=(v[n-1] + v[n-2])^2); v \\ G. C. Greubel, Jun 18 2019

def a(n):
    if (n==0): return 0
    elif (n==1): return 1
    else: return (a(n-1) + a(n-2))^2
[a(n) for n in (0..10)] # G. C. Greubel, Jun 18 2019

a(n+2) = (a(n+1) + a(n))^2. - Benoit Cloitre, Dec 29 2001

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

A076725 a(n) = a(n-1)^2 + a(n-2)^4, a(0) = a(1) = 1.

Original entry on oeis.org

1, 1, 2, 5, 41, 2306, 8143397, 94592167328105, 13345346031444632841427643906, 258159204435047592104207508169153297050209383336364487461

Offset: 0

Michael Somos, Oct 29 2002

a(n) and a(n+1) are relatively prime for n >= 0.
The number of independent sets on a complete binary tree with 2^(n-1)-1 nodes. - Jonathan S. Braunhut (jonbraunhut(AT)usa.net), May 04 2004. For example, when n=3, the complete binary tree with 2 levels has 2^2-1 nodes and has 5 independent sets so a(3)=5. The recursion for number of independent sets splits in two cases, with or without the root node being in the set.
a(10) has 113 digits and is too large to include.

			a(2) = a(1)^2 + a(0)^4 = 1^2 + 1^4 = 2.
a(3) = a(2)^2 + a(1)^4 = 2^2 + 1^4 = 5.
a(4) = a(3)^2 + a(2)^4 = 5^2 + 2^4 = 41.
a(5) = a(4)^2 + a(3)^4 = 41^2 + 5^4 = 2306.
a(6) = a(5)^2 + a(4)^4 = 2306^2 + 41^4 = 8143397.
a(7) = a(6)^2 + a(5)^4 = 8143397^2 + 2306^4 = 94592167328105.

Robert Israel, Table of n, a(n) for n = 0..13
Karl Petersen, Ibrahim Salama, Tree shift complexity, arXiv:1712.02251 [math.DS], 2017.
Index entries for sequences of form a(n+1)=a(n)^2 + ...

Cf. A000283, A005154, A112969, A114793.

A[0]:= 1: A[1]:= 1:
for n from 2 to 10 do
  A[n]:= A[n-1]^2 + A[n-2]^4;
od:
seq(A[i],i=0..10); # Robert Israel, Aug 21 2017

RecurrenceTable[{a[n] == a[n-1]^2 + a[n-2]^4, a[0] ==1, a[1] == 1}, a, {n, 0, 10}] (* Vaclav Kotesovec, Dec 18 2014 *)
NestList[{#[[2]],#[[1]]^4+#[[2]]^2}&,{1,1},10][[All,1]] (* Harvey P. Dale, Jul 03 2021 *)

{a(n) = if( n<2, 1, a(n-1)^2 + a(n-2)^4)}

{a=[0,0];for(n=1,99,iferr(a=[a[2],log(exp(a*[4,0;0,2])*[1,1]~)],E,return([n,exp(a[2]/2^n)])))} \\ To compute an approximation of the constant c1 = exp(lim_{n->oo} (log a(n))/2^n). \\ M. F. Hasler, May 21 2017

a=vector(20); a[1]=1;a[2]=2; for(n=3, #a, a[n]=a[n-1]^2+a[n-2]^4); concat(1, a) \\ Altug Alkan, Apr 04 2018

If b(n) = 1 + 1/b(n-1)^2, b(1)=1, then b(n) = a(n)/a(n-1)^2.
Lim_{n->inf} a(n)/a(n-1)^2 = A092526 (constant).
a(n) is asymptotic to c1^(2^n) * c2.
c1 = 1.2897512927198122075..., c2 = 1/A092526 = A263719 = (1/6)*(108 + 12*sqrt(93))^(1/3) - 2/(108 + 12*sqrt(93))^(1/3) = 0.682327803828019327369483739711... is the root of the equation c2*(1 + c2^2) = 1. - Vaclav Kotesovec, Dec 18 2014

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 15 2007

A112969 a(n) = a(n-1)^4 + a(n-2)^4 for n >= 2 with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 1, 2, 17, 83537, 48698490414981559682, 5624216052381164150697569400035392464306474190030694298257552124199835791859537

Offset: 0

Jonathan Vos Post, Jan 02 2006

A quartic Fibonacci sequence.

This is the quartic (or biquadratic) analog of the Fibonacci sequence similarly to A000283 being the quadratic analog of the Fibonacci sequence. The primes in this sequence begin a(3), a(4), a(5).

			a(3) = 1^4 + 1^4 = 2.
a(4) = 1^4 + 2^4 = 17.
a(5) = 2^4 + 17^4 = 83537.
a(6) = 17^4 + 83537^4 = 48698490414981559682.

Eric Weisstein's World of Mathematics, Quartic Equation.

Cf. A000045, A000283.

RecurrenceTable[{a[1] ==1, a[2] == 1, a[n] == a[n-1]^4 + a[n-2]^4}, a, {n, 1, 8}] (* Vaclav Kotesovec, Dec 18 2014 *)

a(n) ~ c^(4^n), where c = 1.0111288972169538887655499395580320278253918666919181401824606983217263409... . - Vaclav Kotesovec, Dec 18 2014

Name edited by Petros Hadjicostas, Nov 03 2019

a(0)=0 prepended by Alois P. Heinz, Sep 15 2023

A112961 a(n) = a(n-1)^3 + a(n-2)^3 for n >= 2 with a(0) = 0, a(1) = 1.

Original entry on oeis.org

0, 1, 1, 2, 9, 737, 400316282, 64151935432803278787493321, 264015418305763603932856608512044494366944180663171458205345412119783805892929

Offset: 0

Jonathan Vos Post, Jan 02 2006

A cubic Fibonacci sequence.

This is the cubic analog of the Fibonacci sequence analogously to A000283 being the quadratic analog of the Fibonacci sequence.

			a(3) = 1^3 + 1^3 = 2.
a(4) = 1^3 + 2^3 = 9.
a(5) = 2^3 + 9^3 = 737.
a(6) = 9^3 + 737^3 = 400316282.

Cf. A000045, A000283.

a:= proc(n) a(n):= `if`(n<2, n, a(n-1)^3+a(n-2)^3) end:
seq(a(n), n=0..8);  # Alois P. Heinz, Sep 02 2023

RecurrenceTable[{a[1]==a[2]==1,a[n]==a[n-1]^3+a[n-2]^3},a,{n,10}] (* Harvey P. Dale, Aug 24 2014 *)

a(n) ~ b^3^n with b = 1.0275436477.... [Charles R Greathouse IV, Dec 28 2011]

Name edited by Petros Hadjicostas, Nov 03 2019

a(0)=0 prepended by Alois P. Heinz, Sep 02 2023

A112957 a(1) = a(2) = a(3) = 1; for n > 1, a(n+3) = a(n)^2 + a(n+1)^2 + a(n+2)^2.

Original entry on oeis.org

1, 1, 1, 3, 11, 131, 17291, 298995963, 89398586189293211, 7992107212644486930829797919966571, 63873777698404030240264509605345282496735163325301838600463378485931

Offset: 1

Jonathan Vos Post, Jan 02 2006; definition corrected Jan 02 2006

A quadratic tribonacci sequence.

This is to A000283 as a tribonacci (A000213) is to Fibonacci. Two oddities about this sequence: (a) its first 7 terms are identical to terms numbered 2 through 8 of A072878; (b) only one of the first 9 terms are composite. Primes in the sequence begin 3, 11, 131, 17291 and 89398586189293211. What is the next prime?

Seiichi Manyama, Table of n, a(n) for n = 1..14

Cf. A000213, A000283, A072878, A112958, A112959, A112960.

Join[{a=1,b=1,c=1},Table[d=a*a+b*b+c*c;a=b;b=c;c=d,{n,10}]] (* Vladimir Joseph Stephan Orlovsky, Apr 19 2011 *)

A112958 a(1) = a(2) = a(3) = a(4) = 1; for n>1: a(n+4) = a(n)^2 + a(n+1)^2 + a(n+2)^2 + a(n+3)^2.

Original entry on oeis.org

1, 1, 1, 1, 4, 19, 379, 144019, 20741616379, 430214650034342688004, 185084645104171955001009752069374428191659

Offset: 1

Jonathan Vos Post, Jan 02 2006

A quadratic tetranacci sequence.

This is to A000283 as a tetranacci (A000288) is to Fibonacci. Primes in this begin 19, 379.

			1^2 + 4^2 + 19^2 + 379^2 = 144019.

Seiichi Manyama, Table of n, a(n) for n = 1..15

Cf. A000283, A000288, A112957, A112959, A112960.

RecurrenceTable[{a[1] == a[2] == a[3] == a[4] == 1, a[n] == a[n-1]^2 + a[n-2]^2 + a[n-3]^2 + a[n-4]^2}, a, {n, 15}] (* Vincenzo Librandi, Aug 21 2016 *)

A112959 a(1) = a(2) = a(3) = a(4) = a(5) = 1; for n>1: a(n+5) = (a(n))^2 + (a(n+1))^2 + (a(n+2))^2 + (a(n+3))^2 + (a(n+4))^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 5, 29, 869, 756029, 571580604869, 326704387862983487112029, 106735757048926752040856495274871386126283608845

Offset: 1

Jonathan Vos Post, Jan 02 2006

A quadratic pentanacci sequence.

This is to A000283 as a pentanacci (A000322) is to Fibonacci. Primes in this begin a(6) = 5 and a(7) = 29. a(8), a(9), a(10) and a(11) are semiprime.

			5^2 + 29^2 + 869^2 + 756029^2 + 571580604869^2 = 326704387862983487112029.

Seiichi Manyama, Table of n, a(n) for n = 1..16

Cf. A000283, A000288, A112957, A112958, A112960.

RecurrenceTable[{a[1] == a[2] == a[3] == a[4] == a[5] == 1, a[n] == a[n-1]^2 + a[n-2]^2 + a[n-3]^2 + a[n-4]^2 + a[n-5]^2}, a, {n, 16}] (* Vincenzo Librandi, Aug 21 2016 *)

A112960 a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = 1; for n>1: a(n+6) = (a(n))^2 + (a(n+1))^2 + (a(n+2))^2 + (a(n+3))^2 + (a(n+4))^2 + (a(n+5))^2.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 6, 41, 1721, 2963561, 8782696764281, 77135762453320729974211241, 5949925849255124079413733148488788342637650064971321

Offset: 1

Jonathan Vos Post, Jan 02 2006

A quadratic hexanacci sequence.

This is to A000283 as a hexanacci (A000383) is to Fibonacci. Primes in this begin a(8) = 41, a(9) = 1721. Semiprimes begin a(7), a(10), a(12).

			1^2 + 6^2 + 41^2 + 1721^2 + 2963561^2 + 8782696764281^2 = 77135762453320729974211241.

Seiichi Manyama, Table of n, a(n) for n = 1..17

Cf. A000283, A000383, A112957, A112958, A112959.

RecurrenceTable[{a[1] == a[2] == a[3] == a[4] == a[5] == a[6] == 1, a[n] == a[n-1]^2 + a[n-2]^2 + a[n-3]^2 + a[n-4]^2 + a[n-5]^2 + a[n-6]^2}, a, {n, 17}] (* Vincenzo Librandi, Aug 21 2016 *)
nxt[{a_,b_,c_,d_,e_,f_}]:={b,c,d,e,f,a^2+b^2+c^2+d^2+e^2+f^2}; NestList[ nxt,{1,1,1,1,1,1},12][[All,1]] (* Harvey P. Dale, Apr 12 2020 *)

A063827 a(n) = round(sqrt(a(n-2)^2 + a(n-1)^2)) with a(0) = 1 and a(1) = 2.

Original entry on oeis.org

1, 2, 2, 3, 4, 5, 6, 8, 10, 13, 16, 21, 26, 33, 42, 53, 68, 86, 110, 140, 178, 226, 288, 366, 466, 593, 754, 959, 1220, 1552, 1974, 2511, 3194, 4063, 5168, 6574, 8362, 10637, 13530, 17211, 21892, 27847, 35422, 45057, 57314, 72904, 92736, 117962, 150050

Offset: 0

Henry Bottomley, Aug 20 2001

nonn

a(n)/a(n-1) tends towards 1.27201964951... = sqrt((1+sqrt(5))/2). See A139339.

			a(7) = 8 since round(sqrt(5^2 + 6^2)) = round(sqrt(61)) = round(7.8102...) = 8.

Harry J. Smith, Table of n, a(n) for n = 0..500

Cf. A000283, A139339.

nxt[{a_,b_}]:={b,Round[Sqrt[a^2+b^2]]}; NestList[nxt,{1,2},50][[;;,1]] (* Harvey P. Dale, Dec 18 2024 *)

{ default(realprecision, 50); for (n=0, 500, if (n>1, a=round(sqrt(a2^2 + a1^2)); a2=a1; a1=a, if (n, a=a1=2, a=a2=1)); write("b063827.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 01 2009

Missing parenthesis added to definition by Harry J. Smith, Sep 01 2009

cp's OEIS Frontend

A014253 a(n) = b(n)^2, where b(n) = b(n-1)^2 + b(n-2)^2 (A000283).

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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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A076725 a(n) = a(n-1)^2 + a(n-2)^4, a(0) = a(1) = 1.

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A112969 a(n) = a(n-1)^4 + a(n-2)^4 for n >= 2 with a(0) = 0, a(1) = 1.

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A112961 a(n) = a(n-1)^3 + a(n-2)^3 for n >= 2 with a(0) = 0, a(1) = 1.

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A112957 a(1) = a(2) = a(3) = 1; for n > 1, a(n+3) = a(n)^2 + a(n+1)^2 + a(n+2)^2.

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A112958 a(1) = a(2) = a(3) = a(4) = 1; for n>1: a(n+4) = a(n)^2 + a(n+1)^2 + a(n+2)^2 + a(n+3)^2.

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