A000058 -id:A000058 - cp's OEIS frontend

A115590 a(0) = 0; a(n) = (1+a(n-1))^3 for n > 0.

0, 1, 8, 729, 389017000, 58871587162270593034051001, 204040901322752673844230437877671861543858084850895762746141813554591014612008

Offset: 0

Paolo Bonzini, Mar 15 2006; corrected Apr 06 2006 and Jan 19 2007

Take the rooted ternary tree of depth n, with (3^(n+1) - 1) / 2 labeled nodes. Let the number of rooted subtrees be a(n). For example, for n = 1 the a(2) = 8 subtrees are:
 R...R...R...R......R.......R...R.......R
 .../....|....\..../.\...../|...|\...../|\
 ..o.....o.....o..o...o...o.o...o.o...o.o.o
Then a(n+1) = (1+a(n))^3.

Geir Agnarsson, Elie Alhajjar, and Aleyah Dawkins, On locally finite ordered rooted trees and their rooted subtrees, arXiv:2312.11379 [math.CO], 2023.
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 related to rooted trees

Cf. A000578, A004019, A368326, A368327.

{0}~Join~RecurrenceTable[{a[n]==(a[n-1]+1)^3, a[0]==1},a,{n,0,8}] (* Vaclav Kotesovec, May 21 2015 *)

As for A004019, it follows from Aho and Sloane that there is a constant c such that a(n) is the nearest integer to c^(3^n). In fact a(n) = nearest integer to b^(3^n) - 1 where b = 2.0804006677503193521177452323719035237099784935372250879749088464344434056773788...

Name edited by Michael De Vlieger, Dec 21 2023

A139244 a(0) = 4; a(n) = a(n-1)^2 - 1.

Original entry on oeis.org

4, 15, 224, 50175, 2517530624, 6337960442777829375, 40169742574216538983356186036612890624, 1613608218478824775913354216413699241125577233045500390244103887844987109375

Offset: 0

Jonathan Vos Post, Jun 06 2008

This is the next analog of A003096 with different initial value a(0), as starting with a(0) = 2 is A003096 and a(0) = 3 is A003096 with first term omitted. It alternates between even and odd values, specifically between 4 mod 10 and 5 mod 10 and is always composite (by difference of squares factorization).

a(n+2) is divisible by a(n)^2. A007814(a(2 n)) = A153893(n). - Robert Israel, Jul 20 2015

Robert Israel, Table of n, a(n) for n = 0..10
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. A003096, A007814, A153893, A260315.

A[0]:= 4:
for n from 1 to 10 do A[n]:= A[n-1]^2-1 od:
seq(A[i],i=0..10); # Robert Israel, Jul 20 2015

a=4; lst={a}; Do[b=a^2-1; AppendTo[lst,b]; a=b, {n,10}]; lst (* Vladimir Joseph Stephan Orlovsky, Dec 28 2010 *)

a(n)=if(n,a(n-1)^2-1,4) \\ Charles R Greathouse IV, Jul 23 2015

a(n-1) = ceiling(c^(2^n)) where c is a constant between 1 and 2.

More specifically, c=1.9668917617901763653335057202... (sequence A260315). - Chayim Lowen, Jul 17 2015

A166105 Quadratic recurrence from Sylvester's sequence, but starting with a(0)=1 and a(1)=2.

Original entry on oeis.org

1, 2, 4, 14, 184, 33674, 1133904604, 1285739649838492214, 1653126447166808570252515315100129584, 2732827050322355127169206170438813672515557678636778921646668538491883474

Offset: 0

Jaume Oliver Lafont, Oct 06 2009

nonn

a(n) is the size of the set S(n) constructed recursively as follows: Let S(1) = {a,b} and let P(S) be the set of pairs (s,t) where s,t are members of S and s not equal to t. We define S(n+1) as the union of S(n) and P(S(n)). - David M. Cerna, Feb 07 2018

David M. Cerna, Table of n, a(n) for n = 0..12

Cf. A000058.

a:= [1, 2];; for n in [3..13] do a[n]:= a[n-1]^2 - a[n-2]^2 + a[n-2]; od; a; # Muniru A Asiru, Feb 07 2018

a := proc(n) option remember: if n=0 then 1 elif n=1 then 2 elif n>=2 then procname(n-1)^2 - procname(n-2)^2 + procname(n-2) fi; end:
seq(a(n), n = 0..10); # Muniru A Asiru, Feb 07 2018
a:=1:A:=a : to 10 do a:=a*(a-1)+2 : A:=A,a od:
print(A); # Robert FERREOL, May 05 2020

RecurrenceTable[{a[n]==a[n-1]^2-a[n-2]^2+a[n-2],a[0]==1,a[1]==2}, a, {n,0,10}] (* Vaclav Kotesovec, Jan 19 2015 *)

a(n)=if(n<2,[1,2][n+1],a(n-1)^2-a(n-2)^2+a(n-2));

Sum_{n>=0} 1/a(n) = 1.82689305142092757947757234878575... (compare with Sum_{n>=0} 1/A000058(n) = 1).
a(n) ~ c^(2^n), where c = 1.385089248334672909882206535871311526236739234374149506334120193387331772... . - Vaclav Kotesovec, Jan 19 2015
Sum_{n>=1} arctan(1/a(n)) = Pi/4. - Carmine Suriano, Apr 07 2015
a(0)=1, a(n+1) = a(n)*(a(n)-1) + 2. - Robert FERREOL, May 05 2020
a(n) = A002065(n) + 1 = (A232806(n) + 1)/2. - Robert FERREOL, May 31 2020

A343390 a(n) = 1 + Product_{d|n, d < n} a(d).

Original entry on oeis.org

2, 3, 3, 7, 3, 19, 3, 43, 7, 19, 3, 2395, 3, 19, 19, 1807, 3, 2395, 3, 2395, 19, 19, 3, 246546091, 7, 19, 43, 2395, 3, 370387, 3, 3263443, 19, 19, 19, 96124306951, 3, 19, 19, 246546091, 3, 370387, 3, 2395, 2395, 19, 3, 109838449356687381331, 7, 2395, 19, 2395, 3, 246546091, 19

Offset: 1

Ilya Gutkovskiy, Apr 13 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..959

Cf. A000058, A067824, A068334.

a:= proc(n) option remember;
      1+mul(a(d), d=numtheory[divisors](n) minus {n})
    end:
seq(a(n), n=1..55);  # Alois P. Heinz, Apr 13 2021

a[n_] := a[n] = 1 + Product[If[d < n, a[d], 1], {d, Divisors[n]}]; Table[a[n], {n, 55}]

a(p^k) = A000058(k) for p prime.

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

A081466 Consider the mapping f(a/b) = (a^2+b^2)/(a^2-b^2) from rationals to rationals. Starting with 2/1 (a=2, b=1) and applying the mapping to each new (reduced) rational number gives 2/1, 5/3, 17/8, 353/225, ... Sequence gives values of the denominators.

Original entry on oeis.org

1, 3, 8, 225, 36992, 6308330625, 21009822254496776192, 3255818067933293622186199316985612890625, 3264008661830516310447364816658205121507617681188862393654856638929469798612992

Offset: 1

Amarnath Murthy, Mar 22 2003

nonn

Cf. A000058, A081461, A081462, A081465.

{r=2; for(n=1,9,a=numerator(r); b=denominator(r); print1(b,","); r=(a^2+b^2)/(a^2-b^2))}

Edited and extended by Klaus Brockhaus, Mar 24 2003

A110360 Integers with mutual residues of 8.

Original entry on oeis.org

9, 17, 161, 24641, 606981761, 368426853330807041, 135738346255240000293762417728719361, 18424898644107427010977107148874723523180059431182608785043639266493441

Offset: 1

Seppo Mustonen, Sep 04 2005

nonn

This is the special case k=8 of sequences with exact mutual k-residues. In general, a(1)=k+1 and a(n)=min{m | m>a(n-1), mod(m,a(i))=k, i=1,...,n-1}. k=1 gives Sylvester's sequence A000058 and k=2 Fermat sequence A000215.

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!)
Stanislav Drastich, Rapid growth sequences, arXiv:math/0202010 [math.GM], 2002.
S. W. Golomb, On certain nonlinear recurring sequences, Amer. Math. Monthly 70 (1963), 403-405.
S. Mustonen, On integer sequences with mutual k-residues
Index entries for sequences of form a(n+1)=a(n)^2 + ....

Column k=8 of A177888.

RecurrenceTable[{a[1]==9, a[n]==a[n-1]*(a[n-1]-8)+8}, a, {n, 1, 10}] (* Vaclav Kotesovec, Dec 17 2014 *)

a(n) ~ c^(2^n), where c = 1.8813701045812484604409881785833034768479650739052732570542874567824022000... . - Vaclav Kotesovec, Dec 17 2014

A142471 a(0) = a(1) = 0; thereafter a(n) = a(n-1)*a(n-2) + 2.

Original entry on oeis.org

0, 0, 2, 2, 6, 14, 86, 1206, 103718, 125083910, 12973452977382, 1622770224612082123622, 21052933202100473722674133293917606, 34164073141115747076263787631563122725393126176374288934

Offset: 0

N. J. A. Sloane, based on email from Carla J. Garner-Bennett, Nov 13 2008

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..19
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!)

Cf. A000058, A007660, A143684.

I:=[0,0]; [n le 2 select I[n] else Self(n-1)*Self(n-2)+2: n in [1..15]]; // Vincenzo Librandi, Nov 14 2011

a:= proc(n) option remember;
      if n<2 then 0
    else a(n-1)*a(n-2) + 2
      fi; end:
seq(a(n), n=0..15); # G. C. Greubel, Apr 03 2021

a[0] = a[1] = 0; a[n_] := a[n-1] a[n-2] + 2; Table[a[n], {n, 0, 15}] (* T. D. Noe, Nov 14 2011 *)

def a(n): return 0 if n<2 else a(n-1)*a(n-2) + 2
[a(n) for n in (0..15)] # G. C. Greubel, Apr 03 2021

a(n) ~ c^(phi^n), where c = 1.278178162398588325773605473403497130099080978627235683548955136178125... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2015

A143684 a(0) = a(1) = 0; thereafter a(n) = 2a(n-1)a(n-2) + 1.

Original entry on oeis.org

0, 0, 1, 1, 3, 7, 43, 603, 51859, 62541955, 6486726488691, 811385112306041061811, 10526466601050236861337066646958803, 17082036570557873538131893815781561362696563088187144467, 359626974875792367278553795120318710475396935851854517275793126801351587742904492716786003

Offset: 0

N. J. A. Sloane, based on email from Carla J. Garner-Bennett, Nov 13 2008

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..19
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!)

Cf. A000058, A007660, A142471.

I:=[0,0]; [n le 2 select I[n] else 2*Self(n-1)*Self(n-2)+1: n in [1..15]]; // Vincenzo Librandi, Nov 14 2011

a[n_]:=a[n]=If[n<2,0, 2*a[n-1]*a[n-2]+1]; Table[a[n], {n,0,15}] (* G. C. Greubel, May 29 2021 *)

def a(n): return 0 if (n<2) else 2*a(n-1)*a(n-2) + 1
[a(n) for n in (0..10)] # G. C. Greubel, May 29 2021

Equals A142471/2.

a(n) is about 1/2*c^(phi^n), where c = 1.27817816239858832577... and phi is the golden ratio. - Charles R Greathouse IV, Mar 21 2012, corrected by Vaclav Kotesovec, May 05 2015

A177701 Triangle of coefficients of polynomials P_n(z) defined by the recursion P_0(z) = z+1; for n>=1, P_n(z) = z + Product_{k=0..n-1} P_k(z).

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 1, 4, 14, 16, 8, 1, 16, 112, 324, 508, 474, 268, 88, 16, 1, 256, 3584, 22912, 88832, 233936, 443936, 628064, 675456, 557492, 353740, 171644, 62878, 17000, 3264, 416, 32, 1, 65536, 1835008, 24576000, 209715200, 1281482752, 5974786048, 22114709504, 66752724992

Offset: 1

Vladimir Shevelev, Dec 11 2010

Length of the first row is 2; for i>=2, length of the i-th row is 2^{i-2}+1.

			Triangle begins:
   1,   1;
   2,   1;
   2,   4,   1;
   4,  14,  16,   8,   1;
  16, 112, 324, 508, 474, 268, 88, 16, 1;
  ...

Alois P. Heinz, Table of n, a(n) for n = 1..1035
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!)

Cf. A000058, A000215, A000289, A000324, A001543, A001544, A067686, A110360, A000027, A005408, A056220, A177888.

First column gives: A165420.

p:= proc(n) option remember;
       z-> z+ `if`(n=0, 1, p(n-1)(z)*(p(n-1)(z)-z))
    end:
deg:= n-> `if`(n=0, 1, 2^(n-1)):
T:= (n,k)-> coeff(p(n)(z), z, deg(n)-k):
seq(seq(T(n,k), k=0..deg(n)), n=0..6); # Alois P. Heinz, Dec 13 2010

P[0][z_] := z + 1;
P[n_][z_] := P[n][z] = z + Product[P[k][z], {k, 0, n-1}];
row[n_] := CoefficientList[P[n][z], z] // Reverse;
Table[row[n], {n, 0, 6}] // Flatten (* Jean-François Alcover, Jun 11 2018 *)

Another recursion is: P_n(z)=z+P_(n-1)(z)(P_(n-1)(z)-z).

Private values: P_n(0)=1; P_n(-1)=delta_(n,0)-1; {P_n(1)}=A000058; {P_n(2)}=A000215; {P_n(3)}={A000289(n+1)}; {P_n(4)}={A000324(n+1)}; {P_n(5)}={A001543(n+1)}; {P_n(6)}={A001544(n+1)}; {P_n(7)}={A067686(n)}; {P_n(8)}={A110360(n)}; {P_0(n)}={A000027(n+1)}; {P_1(n)}={A005408(n)}; {P_2(n)}={A056220(n+1)}.

More terms from Alois P. Heinz, Dec 13 2010

cp's OEIS Frontend

A115590 a(0) = 0; a(n) = (1+a(n-1))^3 for n > 0.

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A139244 a(0) = 4; a(n) = a(n-1)^2 - 1.

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A166105 Quadratic recurrence from Sylvester's sequence, but starting with a(0)=1 and a(1)=2.

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A343390 a(n) = 1 + Product_{d|n, d < n} a(d).

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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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A081466 Consider the mapping f(a/b) = (a^2+b^2)/(a^2-b^2) from rationals to rationals. Starting with 2/1 (a=2, b=1) and applying the mapping to each new (reduced) rational number gives 2/1, 5/3, 17/8, 353/225, ... Sequence gives values of the denominators.

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A110360 Integers with mutual residues of 8.

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A142471 a(0) = a(1) = 0; thereafter a(n) = a(n-1)*a(n-2) + 2.

A143684 a(0) = a(1) = 0; thereafter a(n) = 2a(n-1)a(n-2) + 1.