A076839 -id:A076839 - cp's OEIS frontend

A135352 Period 5: repeat [1,2,2,1,3].

1, 2, 2, 1, 3, 1, 2, 2, 1, 3, 1, 2, 2, 1, 3, 1, 2, 2, 1, 3, 1, 2, 2, 1, 3, 1, 2, 2, 1, 3, 1, 2, 2, 1, 3, 1, 2, 2, 1, 3, 1, 2, 2, 1, 3, 1, 2, 2, 1, 3, 1, 2, 2, 1, 3, 1, 2, 2, 1, 3, 1, 2, 2, 1, 3, 1, 2, 2, 1, 3, 1, 2, 2, 1, 3, 1, 2, 2, 1, 3, 1, 2, 2, 1, 3, 1, 2, 2, 1, 3, 1, 2, 2, 1, 3

Offset: 1

Roger L. Bagula, Feb 16 2008

This sequence (if extended to be bi-infinite) is the quiddity sequence of the unique width-5 Coxeter frieze pattern A139434; equivalently, if one goes around the (uniquely) triangulated regular pentagon and sequentially looks at its vertices, counting the number of triangles incident with each vertex, then this sequence will be obtained. - Andrey Zabolotskiy, May 04 2023

G. C. Greubel, Table of n, a(n) for n = 1..1000
Karin Baur, Frieze Patterns of Integers, Math. Intelligencer 43, 47-54 (2021). See Example 2 and Figure 4.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A076839, A139434.

Edited by Joerg Arndt, Oct 11 2016

Initial term 1 removed by Joerg Arndt, May 04 2023

A105737 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=4.

Original entry on oeis.org

1, 4, 5, 6, 8, 8, 6, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2

Offset: 1

Zak Seidov, Apr 19 2005

nonn

The sequence depends on seed terms a(1) and a(2); if a(1)=1, a(3)=a(2)+1. All(?) sequences end with cycle={1,2,3,2,1} (or {2,4,6,4,2}, which essentially the same cycle) of length=5. This particular sequence does not merge with A076839 or A105736 and ends with another cycle {2,4,6,4,2}.

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

Cf. A076839, A105736 - A105746.

More terms from Ray Chandler, May 17 2024

A335688 Numerator of b(n), where b(1)=b(2)=1, and for n>2, b(n) = (1+b(n-2))/b(n-1).

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 6, 5, 126, 115, 297108, 1370225, 200590993732752, 531984953824739375, 27113273919595441588692453206842987776, 56653874513318656544104537473266204894369805859375

Offset: 1

N. J. A. Sloane, Jul 17 2020

With a tiny change in subscripts {b(n)} becomes the Lyness 5-cycle, a sequence of period 5, as illustrated in A076839.

			1, 1, 2, 1, 3, 2/3, 6, 5/18, 126/5, 115/2268, 297108/575, 1370225/673840944, 200590993732752/787879375, ...

Cf. A076839, A335689.

A335689 Denominator of b(n), where b(1)=b(2)=1, and for n>2, b(n) = (1+b(n-2))/b(n-1).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 18, 5, 2268, 575, 673840944, 787879375, 135166424574775691397888, 419139972928839518312890625, 3664804294228230129440232448667132628447640073019295936217088, 23745903429826252844483444331595197466499151820665748466452878317718505859375

Offset: 1

N. J. A. Sloane, Jul 17 2020

With a tiny change in subscripts {b(n)} becomes the Lyness 5-cycle, a sequence of period 5, as illustrated in A076839.

			1, 1, 2, 1, 3, 2/3, 6, 5/18, 126/5, 115/2268, 297108/575, 1370225/673840944, 200590993732752/787879375, ...

Cf. A076839, A335688.

nxt[{a_,b_}]:={b,(1+a)/b}; NestList[nxt,{1,1},20][[;;,1]]//Denominator (* Harvey P. Dale, Sep 20 2023 *)

A064268 a(n) = (a(n-1) * a(n-6) + 2 * a(n-3) * a(n-4)) / a(n-7). a(1) = ... = a(7) = 1. Somos-7 variation.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 5, 7, 13, 43, 113, 521, 1241, 3681, 23657, 177721, 679505, 4674203, 27273277, 275517767, 3496390229, 37043734803, 226196947873, 4391322667601, 81041508965617, 1433151398896001, 25397505914206225, 472652420405241521, 9156799134584424289, 499597377081528480243

Offset: 1

Michael Somos, Sep 24 2001

In general, suppose a(n)*a(n-7) = c1*a(n-1)*a(n-6) + c2*a(n-3)*a(n-4) for all n and constants c1,c2. Define u(n) = a(n)*a(n+5)/(a(n+2)*a(n+3)) which satisfies the generalized Lyness recursion u(n) = (c1*u(n-1) + c2)/u(n-2) for all n. For this sequence c1=1, c2=2, u(n) is (1, 1, 3, 5, 7/3, 13/15, 43/35, ...) and satisfies u(n) = (u(n-1) + 2)/u(n-2). See A076839 for Lyness references. - Michael Somos, Sep 26 2022

Harry J. Smith, Table of n, a(n) for n = 1..100
Index entries for two-way infinite sequences

Cf. A006723, A076839.

I:=[1,1,1,1,1,1,1]; [n le 7 select I[n] else (Self(n-1)*Self(n-6) + 2*Self(n-3)*Self(n-4))/Self(n-7): n in [1..30]]; // G. C. Greubel, Feb 21 2018

RecurrenceTable[{a[1]==a[2]==a[3]==a[4]==a[5]==a[6]==a[7]==1,a[n] == (a[n-1]a[n-6]+2a[n-3]a[n-4])/a[n-7]},a,{n,30}] (* Harvey P. Dale, Nov 26 2015 *)

{a(n) = if( n<1, a(8-n), if( n<8, 1, (a(n-1) * a(n-6) + 2 * a(n-3) * a(n-4)) / a(n-7)))};

{ a7=a6=a5=a4=a3=a2=a1=a=1; for (n=1, 100, if (n>7, a=(a1*a6 + 2*a3*a4)/a7; a7=a6; a6=a5; a5=a4; a4=a3; a3=a2; a2=a1; a1=a); write("b064268.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 10 2009

a(8-n) = a(n).

A076824 Let a(1)=a(2)=1, a(n)=(2^ceiling(a(n-1)/2)+1)/a(n-2).

Original entry on oeis.org

1, 1, 3, 5, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 3, 1, 1, 3, 5, 3

Offset: 1

Benoit Cloitre, Nov 24 2002

nonn

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

Cf. A076839.

LinearRecurrence[{0, 0, 0, 0, 1},{1, 1, 3, 5, 3},105] (* Ray Chandler, Aug 25 2015 *)
nxt[{a_,b_}]:={b,(2^Ceiling[b/2]+1)/a}; NestList[nxt,{1,1},110][[;;,1]] (* or *) PadRight[{},120,{1,1,3,5,3}] (* Harvey P. Dale, Jun 09 2025 *)

5-periodic with period (1, 1, 3, 5, 3)

A105745 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1, a(2)=12.

Original entry on oeis.org

1, 12, 13, 4, 9, 9, 4, 5, 6, 8, 8, 6, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4

Offset: 1

Zak Seidov, Apr 19 2005

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

Cf. A076839, A105736 - A105746.

A[1]:= 1: A[2]:= 12:
for n from 3 to 100 do
  R:= map(rhs@op, [msolve(y^2=A[n-1]^2, 4*A[n-2])]);
  ys:= map(t -> (floor((A[n-1]-t)/(4*A[n-2]))+1)*4*A[n-2]+t, R);
  A[n]:= (min(ys)^2-A[n-1]^2)/(4*A[n-2]);
od:
seq(A[i],i=1..100); # Robert Israel, Oct 02 2020

LinearRecurrence[{0,0,0,0,1},{1,12,13,4,9,9,4,5,6,8,8,6,2,4,6,4,2},100] (* or *) PadRight[{1,12,13,4,9,9,4,5,6,8,8,6},100,{4,2,2,4,6}] (* Harvey P. Dale, May 01 2025 *)

a(n)=a(n-5) for n >= 18. - Robert Israel, Oct 02 2020

G.f.: x*(4*x^16 + 4*x^15 + 2*x^14 + 2*x^13 + 3*x^12 - 2*x^11 + x^10 + x^9 - 2*x^8 + 8*x^7 + 8*x^6 - 8*x^5 - 9*x^4 - 4*x^3 - 13*x^2 - 12*x - 1)/(x^5 - 1). - Chai Wah Wu, May 07 2024

More terms from Robert Israel, Oct 02 2020

A105738 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=5.

Original entry on oeis.org

1, 5, 6, 8, 8, 6, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2, 4, 6, 4, 2, 2

Offset: 1

Zak Seidov, Apr 19 2005

nonn

The sequence depends on seed terms a(1) and a(2); if a(1)=1, a(3)=a(2)+1. All(?) sequences end with cycle={1,2,3,2,1} (or {2,4,6,4,2}, which essentially the same cycle) of length = 5. This particular sequence merges with A105737, starting with 2nd term = 5.

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

Cf. A076839, A105736 - A105746.

More terms from Ray Chandler, May 17 2024

A105739 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=6.

Original entry on oeis.org

1, 6, 7, 3, 4, 4, 3, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1

Offset: 1

Zak Seidov, Apr 19 2005

nonn

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

Cf. A076839, A105736 - A105746.

More terms from Ray Chandler, May 17 2024

A105740 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=7.

Original entry on oeis.org

1, 7, 8, 12, 8, 4, 4, 3, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2, 1, 1, 2, 3, 2

Offset: 1

Zak Seidov, Apr 19 2005

nonn

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

Cf. A076839, A105736 - A105746.

More terms from Ray Chandler, May 17 2024

cp's OEIS Frontend

A135352 Period 5: repeat [1,2,2,1,3].

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A105737 For n>2, a(n) > 0 is such that a(n-1)^2+4*a(n-2)*a(n) is a minimal square, a(1)=1,a(2)=4.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A335688 Numerator of b(n), where b(1)=b(2)=1, and for n>2, b(n) = (1+b(n-2))/b(n-1).

Views

Author

Keywords

Comments

Examples

Crossrefs

A335689 Denominator of b(n), where b(1)=b(2)=1, and for n>2, b(n) = (1+b(n-2))/b(n-1).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A064268 a(n) = (a(n-1) * a(n-6) + 2 * a(n-3) * a(n-4)) / a(n-7). a(1) = ... = a(7) = 1. Somos-7 variation.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A076824 Let a(1)=a(2)=1, a(n)=(2^ceiling(a(n-1)/2)+1)/a(n-2).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A105745 For n>2, a(n) > 0 is such that a(n-1)^2+4*a(n-2)*a(n) is a minimal square, a(1)=1, a(2)=12.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A105738 For n>2, a(n) > 0 is such that a(n-1)^2+4*a(n-2)*a(n) is a minimal square, a(1)=1,a(2)=5.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A105739 For n>2, a(n) > 0 is such that a(n-1)^2+4*a(n-2)*a(n) is a minimal square, a(1)=1,a(2)=6.

Views

Author

Keywords

Links

Crossrefs

A105737 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=4.

A105745 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1, a(2)=12.

A105738 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=5.

A105739 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=6.

A105740 For n>2, a(n) > 0 is such that a(n-1)^2+4a(n-2)a(n) is a minimal square, a(1)=1,a(2)=7.