A076839 -id:A076839 - cp's OEIS frontend

A105741 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)=8.

1, 8, 9, 14, 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

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

A105742 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)=9.

Original entry on oeis.org

1, 9, 10, 16, 8, 3, 5, 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, 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

A105743 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)=10.

Original entry on oeis.org

1, 10, 11, 6, 5, 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

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

A105744 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)=11.

Original entry on oeis.org

1, 11, 12, 20, 8, 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

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

A111340 Number of positive integer 2-friezes with n-1 nontrivial rows.

Original entry on oeis.org

1, 5, 51, 868, 26952

Offset: 1

N. J. A. Sloane, based on correspondence from James Propp, May 08 2005

The n-th term is the number of positive integer tables a(i,m) (with i running from 1 to n+3 and m running from minus infinity to infinity) subject to the boundary conditions a(i,m) = 0 when i = 1 or i = n+3 and a(i,m) = 1 when i = 2 or i = n+2 and the internal condition a(i,m-1) a(i,m+1) = a(i-1,m) a(i+1,m) + a(i,m) when i is strictly between 2 and n+2.
It is not known as of this writing whether any or all of the terms of the sequence beyond 868 are finite. If the final term "a(i,n)" in the internal condition is replaced by "1", then what we are looking is just a frieze pattern a la Conway and Coxeter (or rather two interlaced frieze patterns that do not interact at all).
According to the lecture notes by S. Morier-Genoud (see paragraph "2-frieze of positive integers"), a(5) is conjectured to be 26952, and it is proved that there are no more finite terms. - Andrei Zabolotskii, Nov 01 2022

			The number 1 in the sequence is counting the rather boring configuration
    0 0 0 0 0 0 0 0
... 1 1 1 1 1 1 1 1 ...
    1 1 1 1 1 1 1 1
    0 0 0 0 0 0 0 0
The number 5 is counting the configuration
    0 0 0 0 0 0 0 0 0 0
    1 1 1 1 1 1 1 1 1 1
... 1 1 2 3 2 1 1 2 3 2 ...
    1 1 1 1 1 1 1 1 1 1
    0 0 0 0 0 0 0 0 0 0
and its four distinct cyclic shifts, each of which repeats with period 5 (note the Lyness 5-cycle A076839 in the middle).
a(2) = A000108(3) = number of friezes of type A_2 (cyclic shifts of A139434), a(3) = A247415(4). a(4) and a(5) also count friezes of types resp. E_6 and E_8.

Sophie Morier-Genoud, Lecture notes on Integrable Systems and Friezes, 2017.
Robin Zhang, A positive Siegel theorem: Dynkin friezes and positive Mordell-Schinzel, arXiv:2503.08800 [math.NT], 2025.

Cf. A000108, A247415, A076839, A139434.

The last finite term, a(5), added based on Zhang's preprint and name clarified by Andrei Zabolotskii, May 14 2025

A335690 a(1) = 1, a(2) = a(3) = 2; a(n) = (a(n-1) + a(n-2) + 1)/a(n-3) (for n>3).

Original entry on oeis.org

1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4, 5, 2, 2, 1, 2, 2, 5, 4

Offset: 1

N. J. A. Sloane, Jul 19 2020

nonn

This is another illustration of the 8-cycle discovered by H. Todd - see Lyness, Note 1847. Compare A076844. - N. J. A. Sloane, Jul 19 2020

R. C. Lyness, Note 1581. Cycles, Math. Gazette, 26 (1942), 62.
R. C. Lyness, Note 1847. Cycles, Math. Gaz., 29 (1945), 231-233.
R. C. Lyness, Note 2952. Cycles, Math. Gaz., 45 (1961), 207-209.

Cf. A076839, A076844.

a := 1; b := 1; c := 1; f := proc(n) option remember; global a,b,c; if n=1 then RETURN(a); fi; if n=2 then RETURN(b); fi; if n=3 then RETURN(c); fi; RETURN((f(n-1)+f(n-2)+1)/f(n-3)); end;

RecurrenceTable[{a[1]==1,a[2]==a[3]==2,a[n]==(a[n-1]+a[n-2]+1)/a[n-3]},a,{n,90}] (* or *) PadRight[{},90,{1,2,2,5,4,5,2,2}] (* Harvey P. Dale, May 28 2021 *)

A351916 a(1) = a(2) = 1; for n >= 2, a(n+1) = (a(n)^7 + 1)/a(n-1).

Original entry on oeis.org

1, 1, 2, 129, 297233651245505, 1588898389043626055434220300433167237829218942966252641093888571632886068535351219199489258571766594

Offset: 1

Robert Dougherty-Bliss, Feb 25 2022

nonn

a(7) has 680 digits.

Cf. A076839, A003819, A003818, A003821.

a:=proc(n) option remember: if n <= 2 then 1: else (a(n-1)^7+1)/a(n-2): fi: end:

cp's OEIS Frontend

A105741 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)=8.

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A105742 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)=9.

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A105743 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)=10.

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A105744 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)=11.

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Extensions

A111340 Number of positive integer 2-friezes with n-1 nontrivial rows.

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A335690 a(1) = 1, a(2) = a(3) = 2; a(n) = (a(n-1) + a(n-2) + 1)/a(n-3) (for n>3).

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A351916 a(1) = a(2) = 1; for n >= 2, a(n+1) = (a(n)^7 + 1)/a(n-1).

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Maple

A105741 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)=8.

A105742 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)=9.

A105743 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)=10.

A105744 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)=11.