A064526 -id:A064526 - cp's OEIS frontend

A064183 Define a pair of sequences by p(0) = 0, q(0) = p(1) = q(1) = 1, q(n+1) = p(n)*q(n-1), p(n+1) = q(n+1) + q(n) for n > 0; then a(n) = q(n) and A064526(n) = p(n).

1, 1, 1, 2, 3, 10, 39, 490, 20631, 10349290, 213941840151, 2214253254659846890, 473721461633379426414550183191, 1048939288228833100615882755549676600679754298090

Offset: 0

Michael Somos, Sep 20 2001

Michael Somos and R. Haas, A linked pair of sequences implies the primes are infinite, Amer. Math. Monthly, 110(6) (2003), 539-540.

Cf. A001622, A003686, A064526, A064847, A070231, A070233, A070234, A094303, A236394.

Flatten[{1, RecurrenceTable[{a[n]==(a[n-1]+a[n-2])*a[n-2], a[1]==1, a[2]==1},a,{n,1,10}]}] (* Vaclav Kotesovec, May 21 2015 *)

{a(n) = local(v); if( n<3, n>=0, v = [1,1]; for( k=3, n, v = [v[2], v[1] * (v[1] + v[2])]); v[2])}

{a(n) = if( n<3, n>=0, (a(n-1) + a(n-2)) * a(n-2))}

a(n) = (a(n-1) + a(n-2))*a(n-2) for n >= 2.
Lim_{n -> infinity} a(n)/a(n-1)^phi = 1, where phi = A001622. - Gerald McGarvey, Aug 29 2004
a(n) ~ c^(phi^n), where c = 1.23642417842410860616065684299168229758826316461949675490684055924721259... and phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2015

A236394 Primes produced by A064526, in order of appearance.

Original entry on oeis.org

2, 3, 5, 13, 7, 23, 21121, 853, 12157, 213952189441, 31, 71427531245215711, 17, 163, 68743, 28031803, 88717035481559, 4549, 38197, 3835420378661, 1573954557128833179852821957

Offset: 1

N. J. A. Sloane, Jan 28 2014

Cf. A064526, A064183.

A003686 Number of genealogical 1-2 rooted trees of height n.

Original entry on oeis.org

1, 2, 3, 5, 11, 41, 371, 13901, 5033531, 69782910161, 351229174914190691, 24509789089655802510792656021, 8608552999157278575508415639286249242844899051

Offset: 1

Vsevolod F. Lev, c. 1998

Let u(n), v(n) be defined by u(1) = v(1) = 1, u(n+1) = u(n) + v(n) and v(n+1) = u(n)*v(n) for n >= 1; then a(n) = u(n) and A064847(n) = v(n). - Benoit Cloitre, Apr 01 2002 [Edited by Petros Hadjicostas, May 11 2020]
Consider the mapping f(a/b) = (a + b)/(a*b). Taking a = 1 and b = 1 to start with and carrying out this mapping repeatedly on each new (reduced) rational number gives the following sequence 1/1, 2/1, 3/2, 5/6, 11/30, ... The current sequence contains the numerators. - Amarnath Murthy, Mar 24 2003
An infinite coprime sequence defined by recursion. - Michael Somos, Mar 19 2004

D. Parisse, The Tower of Hanoi and the Stern-Brocot Array, Thesis, Munich, 1997.

Franklin T. Adams-Watters, Table of n, a(n) for n = 1..19
Index entries for sequences related to Stern's sequences
Index entries for sequences related to rooted trees

Cf. A001622, A001685, A064526, A064847, A070231, A070233, A070234, A094303.

I:=[1,2]; [n le 2 select I[n] else Self(n-1)+Self(n-2)*(Self(n-1)-Self(n-2)): n in [1..14]]; // Vincenzo Librandi, Jul 19 2016

RecurrenceTable[{a[1]==1, a[2]==2, a[n]==a[n-1]+a[n-2](a[n-1]-a[n-2])}, a[n],{n,15}] (* Harvey P. Dale, Jul 27 2011 *)
Re[NestList[Re@#+(1+I Re@#)Im@#&, 1+I, 15]] (* Vladimir Reshetnikov, Jul 18 2016 *)

a(n) = local(an); if(n<1, 0, an=vector(max(2,n)); an[1]=1; an[2]=2; for(k=3, n, an[k]=an[k-1] - an[k-2]^2 + an[k-1]*an[k-2]); an[n])

Limit_{n -> infinity} a(n)^phi/A064847(n) = 1, where phi = (1 + sqrt(5))/2 is the golden ratio. - Benoit Cloitre, May 08 2002
Numerator of b(n), where b(n) = 1/numerator(b(n-1)) + 1/denominator(b(n-1)) for n >= 2 with b(1) = 1.
a(n+1) = a(n) + a(1)*a(2)*...*a(n-1) for n >= 2. Also a(n+1) = a(n) + a(n-1)*(a(n) - a(n-1)) for n >= 2. In both cases, we start with a(1) = 1 and a(2) = 2.
a(n) ~ c^(phi^n), where c = 1.22508584062304325811405322247537613534139348463831009881946422737141574647... and phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, May 21 2015

Additional description from Andreas M. Hinz and Daniele Parisse

A001685 a(0) = 1, a(1) = 2, a(2) = 3; for n >= 3, a(n) = a(n-2) + a(n-1)*Product_{i=1..n-3} a(i).

Original entry on oeis.org

1, 2, 3, 5, 13, 83, 2503, 976253, 31601312113, 2560404986164794683, 202523113189037952478722304798003, 506227391211661106785411233681995783881012463859772443053

Offset: 0

N. J. A. Sloane

nonn

From a continued fraction.

Every term is relatively prime to all others. - Michael Somos, Feb 01 2004

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).

John Cerkan, Table of n, a(n) for n = 0..16
V. C. Harris, Another proof of the infinitude of primes, Amer. Math. Monthly, 63 (1956), 711.
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012. - From _N. J. A. Sloane_, Jun 13 2012

Cf. A003686, A064526, A109134.

Clear[a]; a[0]=1; a[1]=2; a[2]=3; a[n_]:=a[n] = a[n-2] + a[n-1]*Product[a[j],{j,1,n-3}]; Table[a[n],{n,0,15}] (* Vaclav Kotesovec, May 21 2015 *)
Clear[a];RecurrenceTable[{a[n]==a[n-2]+a[n-1]*a[n-3]*(a[n-1]-a[n-3])/a[n-2],a[0]==1,a[1]==2,a[2]==3},a,{n,0,15}] (* Vaclav Kotesovec, May 21 2015 *)

a(n)=if(n<3,max(0,n+1),a(n-2)+a(n-1)*prod(i=1,n-3,a(i))) /* Michael Somos, Feb 01 2004 */

a(n) = a(n-2) + a(n-1)*a(n-3)*(a(n-1)-a(n-3))/a(n-2). - Vaclav Kotesovec, May 21 2015

a(n) ~ c^(d^n), where d = A109134 = 1.754877666246692760049508896358528691894606617772793143989283970646... is the root of the equation d*(d-1)^2 = 1, c = 1.3081335128180696870655208993764956995000211962454918672885690026423582299... . - Vaclav Kotesovec, May 21 2015

Edited by N. J. A. Sloane, Jun 12 2006

A070231 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)v(k) + v(k)w(k) + w(k)u(k), and w(k+1) = u(k)v(k)*w(k) for k >= 1; then a(n) = u(n).

Original entry on oeis.org

1, 3, 7, 31, 1279, 9202687, 3692849258577919, 98367959484921734629696721986125823, 3882894052327310957045599009145809243674851356642054390303168725061781159935999

Offset: 1

Benoit Cloitre, May 08 2002

Petros Hadjicostas, Table of n, a(n) for n = 1..12

Cf. A003686, A064183, A064526, A064847, A070233 (= w), A070234 (= v), A094303, A160389, A255249.

a[1] = 1; v[1] = 1; w[1] = 1; a[k_] := a[k] = a[k - 1] + v[k - 1] + w[k - 1]; v[k_] := v[k] = a[k - 1]*v[k - 1] + v[k - 1]*w[k - 1] + w[k - 1]*a[k - 1]; w[k_] := w[k] = a[k - 1]*v[k - 1]*w[k - 1]; Table[a[n], {n, 1, 9}] (* Vaclav Kotesovec, May 11 2020 *)

lista(nn) = {my(u = vector(nn)); my(v = vector(nn)); my(w = vector(nn)); u[1] = 1; v[1] = 1; w[1] = 1; for (n=2, nn, u[n] = u[n-1] + v[n-1] + w[n-1]; v[n] = u[n-1]*v[n-1] + v[n-1]*w[n-1] + w[n-1]*u[n-1]; w[n] = u[n-1]*v[n-1]*w[n-1];); u; } \\ Petros Hadjicostas, May 11 2020

Let C be the positive root of x^3 + x^2 - 2*x - 1 = 0; that is, C = 1.246979603717... = A255249. Then Lim_{n -> infinity} u(n)^(C+1)/w(n)= Lim_{n -> infinity} v(n)^C/w(n) = Lim_{n -> infinity} u(n)^B/v(n) = 1 with B = C + 1 - 1/(1 + C) = 1.8019377... = A160389. [corrected by Vaclav Kotesovec, May 11 2020]

a(n) ~ gu^((1 + C)^n), where C is defined above and gu = 1.131945853718244297... The relation between constants gu, gv (see A070234) and gw (see A070233) is gu^(1 + C) = gv^C = gw. - Vaclav Kotesovec, May 11 2020

A070233 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)v(k) + v(k)w(k) + w(k)u(k), and w(k+1) = u(k)v(k)*w(k) for k >= 1; then a(n) = w(n).

Original entry on oeis.org

1, 1, 9, 945, 8876385, 3689952451492545, 98367948795841301790914258556831105, 3882894052327309905582682317031276840071039865528864289025562807872336355445505

Offset: 1

Benoit Cloitre, May 08 2002

Next term is too large to include.

Petros Hadjicostas, Table of n, a(n) for n = 1..11

Cf. A003686, A064183, A064526, A064847, A070231 (= u), A070234 (= v), A094303, A160389, A255249.

u[1] = 1; v[1] = 1; a[1] = 1; u[k_] := u[k] = u[k - 1] + v[k - 1] + a[k - 1]; v[k_] := v[k] = u[k - 1]*v[k - 1] + v[k - 1]*a[k - 1] + a[k - 1]*u[k - 1]; a[k_] := a[k] = u[k - 1]*v[k - 1]*a[k - 1]; Table[a[n], {n, 1, 9}] (* Vaclav Kotesovec, May 11 2020 *)

lista(nn) = {my(u = vector(nn)); my(v = vector(nn)); my(w = vector(nn)); u[1] = 1; v[1] = 1; w[1] = 1; for (n=2, nn, u[n] = u[n-1] + v[n-1] + w[n-1]; v[n] = u[n-1]*v[n-1] + v[n-1]*w[n-1] + w[n-1]*u[n-1]; w[n] = u[n-1]*v[n-1]*w[n-1]; ); w; } \\ Petros Hadjicostas, May 11 2020

Let C be the positive root of x^3 + x^2 - 2*x - 1 = 0: that is, C = 1.246979603717... = A255249. Then Lim_{n -> infinity} u(n)^(C+1)/w(n)= Lim_{n -> infinity} v(n)^C/w(n) = Lim_{n -> infinity} u(n)^B/v(n) = 1 with B = C + 1 - 1/(1 + C) = 1.8019377... = A160389. [corrected by Vaclav Kotesovec, May 11 2020]

a(n) ~ gw^((C + 1)^n), where C is defined above and gw = 1.321128752475732548... The relation between constants gu (see A070231), gv (see A070234) and gw is gu^(1 + C) = gv^C = gw. - Vaclav Kotesovec, May 11 2020

A070234 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)v(k) + v(k)w(k) + w(k)u(k), and w(k+1) = u(k)v(k)*w(k); then a(n) = v(n).

Original entry on oeis.org

1, 3, 15, 303, 325023, 2896797882687, 10689080432835089614170716799, 1051462916692114532403603811392745230616355871287492722818364671

Offset: 1

Benoit Cloitre, May 08 2002

Petros Hadjicostas, Table of n, a(n) for n = 1..11

Cf. A003686, A064183, A064526, A064847, A070231 (= u), A070233 (= w), A094303, A160389, A255249.

u[1] = 1; a[1] = 1; w[1] = 1; u[k_] := u[k] = u[k - 1] + a[k - 1] + w[k - 1]; a[k_] := a[k] = u[k - 1]*a[k - 1] + a[k - 1]*w[k - 1] + w[k - 1]*u[k - 1]; w[k_] := w[k] = u[k - 1]*a[k - 1]*w[k - 1]; Table[a[n], {n, 1, 9}] (* Vaclav Kotesovec, May 11 2020 *)

lista(nn) = {my(u = vector(nn)); my(v = vector(nn)); my(w = vector(nn)); u[1] = 1; v[1] = 1; w[1] = 1; for (n=2, nn, u[n] = u[n-1] + v[n-1] + w[n-1]; v[n] = u[n-1]*v[n-1] + v[n-1]*w[n-1] + w[n-1]*u[n-1]; w[n] = u[n-1]*v[n-1]*w[n-1]; ); v; } \\ _Petros Hadjicostas, May 11 2020

Let C be the positive root of x^3 + x^2 - 2*x - 1 = 0; that is, C = 1.246979603717... = A255249. Then Lim_{n -> infinity} u(n)^(C+1)/w(n) = Lim_{n -> infinity} v(n)^C/w(n) = Lim_{n -> infinity} u(n)^B/v(n) = 1 with B = C + 1 - 1/(1 + C) = 1.8019377... = A160389. [corrected by Vaclav Kotesovec, May 11 2020]

a(n) ~ gv^((C + 1)^n), where C is defined above and gv = 1.250231610564761084... The relation between constants gu (see A070231), gv and gw (see A070233) is gu^(1 + C) = gv^C = gw. - Vaclav Kotesovec, May 11 2020

A094303 a(1) = 1, a(2) = 2, and a(n+1) = a(n) * sum of all previous terms up to a(n-1) for n >= 2.

Original entry on oeis.org

1, 2, 2, 6, 30, 330, 13530, 5019630, 69777876630, 351229105131280530, 24509789089304573335878465330, 8608552999157278550998626549630446732052243030

Offset: 1

Amarnath Murthy, Apr 29 2004

nonn

From Petros Hadjicostas, May 11 2020: (Start)
R. J. Mathar's conjecture is correct and this is identical to A064847 starting at n = 3. To see why this is the case, consider the sequences u(n) and v(n) defined by u(1) = v(1) = 1, and u(k+1) = u(k) + v(k), v(k+1) = u(k)*v(k) for k >= 1. Then u(n) = A003686(n) and v(n) = A064847(n) for n >= 1.
Then v(n) = u(n+1) - u(n), and thus Sum_{k=1..n-1} v(k) = u(n) - u(1) = u(n) - 1 for n >= 2. Then v(n-1) + ... + v(3) + (v(2) + 1) + v(1) = u(n) for n >= 3, and hence v(n)*(v(n-1) + ... + v(3) + (v(2) + 1) + v(1)) = u(n)*v(n) = v(n+1).
Since v(1) = 1 = a(1) and v(2) + 1 = 2 = a(2), the sequence (v(1), v(2) + 1, v(3), ..., v(n), ...) is identical to the current sequence. Hence, a(n) = v(n) = u(n+1) - u(n) = A003686(n+1) - A003686(n) for n >= 3. (End)

Petros Hadjicostas, Table of n, a(n) for n = 1..17

See A064847 for another version.

Cf. A003686, A064183, A064526, A070231, A070233, A070234.

nxt[{t1_,t2_,a_}]:=Module[{c=t1*a},{t1+t2,c,c}]; Join[{1},NestList[nxt,{1,2,2},10][[All,2]]] (* Harvey P. Dale, Aug 30 2020 *)

lista(nn) = { my(va = vector(nn)); va[1] = 1; va[2] = 2; for(n=3, nn, va[n] = va[n-1]*sum(k=1, n-2, va[k]);); va; } \\ Petros Hadjicostas, May 11 2020

Conjecture: a(n) = A003686(n+1) - A003686(n) for n >= 3. - R. J. Mathar, Apr 24 2007

More terms from Gareth McCaughan, Jun 10 2004

A002715 An infinite coprime sequence defined by recursion.

Original entry on oeis.org

3, 7, 23, 47, 1103, 2207, 2435423, 4870847, 11862575248703, 23725150497407, 281441383062305809756861823, 562882766124611619513723647, 158418504200047111075388369241884118003210485743490303

Offset: 0

N. J. A. Sloane

nonn

Every term is relatively prime to all others.

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).

Vincenzo Librandi, Table of n, a(n) for n = 0..21
A. W. F. Edwards, Infinite coprime sequences, Math. Gaz., 48 (1964), 416-422.
A. W. F. Edwards, Infinite coprime sequences, Math. Gaz., 48 (1964), 416-422. [Annotated scanned copy]
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018.

Cf. A001685, A003686, A064526.

a[n_?OddQ] := a[n] = 2*a[n-1] + 1; a[n_?EvenQ] := a[n] = (a[n-1]^2 - 3)/2; a[0] = 3; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Jan 25 2013 *)

a(n)=if(n<1,3*(n==0),if(n%2,2*a(n-1)+1,(a(n-1)^2-3)/2))

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

More terms from Jeffrey Shallit

Edited by Michael Somos, Feb 01 2004

A002716 An infinite coprime sequence defined by recursion.

Original entry on oeis.org

3, 5, 13, 17, 241, 257, 65281, 65537, 4294901761, 4294967297, 18446744069414584321, 18446744073709551617, 340282366920938463444927863358058659841

Offset: 0

N. J. A. Sloane

nonn

Every term is relatively prime to all others. - Michael Somos, Feb 01 2004

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).

A. W. F. Edwards, Infinite coprime sequences, Math. Gaz., 48 (1964), 416-422.
A. W. F. Edwards, Infinite coprime sequences, Math. Gaz., 48 (1964), 416-422. [Annotated scanned copy]
R. Mestrovic, Euclid's theorem on the infinitude of primes: a historical survey of its proofs (300 BC--2012) and another new proof, arXiv preprint arXiv:1202.3670 [math.HO], 2012-2018.

Cf. A000215, A001685, A002715, A003686, A064526, A220294.

a[0] = 3; a[1] = 5;
a[n_] := a[n] = If[OddQ[n], a[n-1] + a[n-2] - 1, a[n-1]^2 - 3*a[n-1] + 3];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Aug 16 2018, after Michel Somos *)

{a(n) = if( n<2, 3 * (n>=0) + 2 * (n>0), if( n%2, a(n-1) + a(n-2) - 1, a(n-1)^2 - 3 * a(n-1) + 3))} /* Michael Somos, Feb 01 2004 */

a(2*n + 1) = a(2*n) + a(2*n - 1) - 1, a(2*n) = a(2*n - 1)^2 - 3 * a(2*n - 1) + 3, a(0) = 3, a(1) = 5. - Michael Somos, Feb 01 2004
Conjecture: a(2n+1)=A001146(n+1)+1. - R. J. Mathar, May 15 2007
a(2*n) = A220294(n). a(2*n + 1) = A000215(n+1). - Michael Somos, Dec 10 2012

More terms from Jeffrey Shallit

Edited by Michael Somos, Feb 01 2004

cp's OEIS Frontend

A064183 Define a pair of sequences by p(0) = 0, q(0) = p(1) = q(1) = 1, q(n+1) = p(n)*q(n-1), p(n+1) = q(n+1) + q(n) for n > 0; then a(n) = q(n) and A064526(n) = p(n).

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A236394 Primes produced by A064526, in order of appearance.

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A003686 Number of genealogical 1-2 rooted trees of height n.

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A001685 a(0) = 1, a(1) = 2, a(2) = 3; for n >= 3, a(n) = a(n-2) + a(n-1)*Product_{i=1..n-3} a(i).

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A070231 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)*v(k) + v(k)*w(k) + w(k)*u(k), and w(k+1) = u(k)*v(k)*w(k) for k >= 1; then a(n) = u(n).

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A070233 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)*v(k) + v(k)*w(k) + w(k)*u(k), and w(k+1) = u(k)*v(k)*w(k) for k >= 1; then a(n) = w(n).

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A070234 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)*v(k) + v(k)*w(k) + w(k)*u(k), and w(k+1) = u(k)*v(k)*w(k); then a(n) = v(n).

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A094303 a(1) = 1, a(2) = 2, and a(n+1) = a(n) * sum of all previous terms up to a(n-1) for n >= 2.

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A070231 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)v(k) + v(k)w(k) + w(k)u(k), and w(k+1) = u(k)v(k)*w(k) for k >= 1; then a(n) = u(n).

A070233 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)v(k) + v(k)w(k) + w(k)u(k), and w(k+1) = u(k)v(k)*w(k) for k >= 1; then a(n) = w(n).

A070234 Let u(k), v(k), w(k) satisfy the recursions u(1) = v(1) = w(1) = 1, u(k+1) = u(k) + v(k) + w(k), v(k+1) = u(k)v(k) + v(k)w(k) + w(k)u(k), and w(k+1) = u(k)v(k)*w(k); then a(n) = v(n).