A010685 -id:A010685 - cp's OEIS frontend

A040005 Continued fraction for sqrt(8).

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

Offset: 0

N. J. A. Sloane

			2.828427124746190097603377448... = 2 + 1/(1 + 1/(4 + 1/(1 + 1/(4 + ...)))). - _Harry J. Smith_, Jun 02 2009

James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 276.

Harry J. Smith, Table of n, a(n) for n = 0..20000
G. Xiao, Contfrac.
Index entries for continued fractions for constants.
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A010466 (decimal expansion).

Essentially the same as A010685.

Digits := 100: convert(evalf(sqrt(N)),confrac,90,'cvgts'):

ContinuedFraction[Sqrt[8],300] (* Vladimir Joseph Stephan Orlovsky, Mar 04 2011 *)

{ allocatemem(932245000); default(realprecision, 16000); x=contfrac(sqrt(8)); for (n=0, 20000, write("b040005.txt", n, " ", x[n+1])); } \\ Harry J. Smith, Jun 02 2009

From Amiram Eldar, Nov 12 2023: (Start)
Multiplicative with a(2^e) = 4, and a(p^e) = 1 for an odd prime p.
Dirichlet g.f.: zeta(s) * (1 + 3/2^s). (End)
G.f.: (2 + x + 2*x^2)/(1 - x^2). - Stefano Spezia, Jul 26 2025

A094015 Expansion of (1+4x)/(1-8x^2).

Original entry on oeis.org

1, 4, 8, 32, 64, 256, 512, 2048, 4096, 16384, 32768, 131072, 262144, 1048576, 2097152, 8388608, 16777216, 67108864, 134217728, 536870912, 1073741824, 4294967296, 8589934592, 34359738368, 68719476736, 274877906944

Offset: 0

Paul Barry, Apr 21 2004

Row sums of triangle A135838. - Gary W. Adamson, Dec 01 2007

Row sums of triangle A152842. - Reinhard Zumkeller, May 01 2014

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000
Index to divisibility sequences
Index entries for linear recurrences with constant coefficients, signature (0,8).

Cf. A010685, A094014, A135838, A152842.

a094015 = sum . a152842_row  -- Reinhard Zumkeller, May 01 2014

[2*8^Floor((n-1)/2)*(3+(-1)^n): n in [0..30]]; // G. C. Greubel, Nov 22 2021

a:=n->mul(3-(-1)^j,j=1..n):seq(a(n),n=0..25); # Zerinvary Lajos, Dec 13 2008

Table[8^Floor[n/2]*Mod[4^n, 5], {n, 0, 30}] (* G. C. Greubel, Nov 22 2021 *)

[8^(n//2)*(4^n%5) for n in (0..30)] # G. C. Greubel, Nov 22 2021

a(n) = 2^(3*n/2)*(1 + sqrt(2) + (-1)^n*(1 - sqrt(2)))/2.
a(n) = (1/4)*(3 + (-1)^n)*8^floor((n+1)/2). - Paul Barry, Jul 14 2004
a(n) = (1 + sqrt(2))*(2*sqrt(2))^n/2 + (1 - sqrt(2))*(-2*sqrt(2))^n/2. Third binomial transform is A002315 (NSW numbers). - Paul Barry, Jul 17 2004
a(n) = 2^A007494(n). - Paul Barry, Aug 18 2007
a(n) = A016116(n+1)*A000079(n). - R. J. Mathar, Jul 08 2009
a(n+3) = a(n+2)*a(n+1)/a(n). - Reinhard Zumkeller, Mar 04 2011
a(n) = 8^floor(n/2)*A010685(n). - G. C. Greubel, Nov 22 2021

A010698 Period 2: repeat (2,8).

Original entry on oeis.org

2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2, 8, 2

Offset: 0

N. J. A. Sloane

This is the regular (simple) continued fraction for (2+sqrt(5))/2 = A176055. - Antonia Redondo Buitrago, Jul 30 2009

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

A010698:=n->5-3*(-1)^n; seq(A010698(n), n=0..100); # Wesley Ivan Hurt, Mar 26 2014

Table[5-3(-1)^n, {n, 0, 100}] (* Wesley Ivan Hurt, Mar 26 2014 *)
PadRight[{},120,{2,8}] (* Harvey P. Dale, Oct 31 2016 *)

A010698(n):=if evenp(n) then 2 else 8$
makelist(A010698(n),n,0,30); /* Martin Ettl, Nov 09 2012 */

a(n)=n%2*6+2 \\ Charles R Greathouse IV, Jun 11 2015

a(n) = -3*(-1)^n+5. - Paolo P. Lava, Oct 20 2006
G.f.: 2(1+4x)/((1-x)(1+x)). a(n) = 2*A010685(n). - R. J. Mathar, Oct 20 2008
a(n) = (A010674(n)+1)*2. - Martin Ettl, Nov 09 2012

A274913 Square array read by antidiagonals upwards in which each new term is the least positive integer distinct from its neighbors.

Original entry on oeis.org

1, 2, 3, 1, 4, 1, 2, 3, 2, 3, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 1, 4, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 2, 3, 1, 4, 1, 4, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3

Offset: 1

Omar E. Pol, Jul 11 2016

This is also a triangle read by rows in which each new term is the least positive integer distinct from its neighbors.
In the square array we have that:
Antidiagonal sums give the positive terms of A008851.
Odd-indexed rows give A010684.
Even-indexed rows give A010694.
Odd-indexed columns give A000034.
Even-indexed columns give A010702.
Odd-indexed antidiagonals give the initial terms of A010685.
Even-indexed antidiagonals give the initial terms of A010693.
Main diagonal gives A010685.
This is also a triangle read by rows in which each new term is the least positive integer distinct from its neighbors.
In the triangle we have that:
Row sums give the positive terms of A008851.
Odd-indexed columns give A000034.
Even-indexed columns give A010702.
Odd-indexed diagonals give A010684.
Even-indexed diagonals give A010694.
Odd-indexed rows give the initial terms of A010685.
Even-indexed rows give the initial terms of A010693.
Odd-indexed antidiagonals give the initial terms of A010684.
Even-indexed antidiagonals give the initial terms of A010694.

			The corner of the square array begins:
1, 3, 1, 3, 1, 3, 1, 3, 1, 3, ...
2, 4, 2, 4, 2, 4, 2, 4, 2, ...
1, 3, 1, 3, 1, 3, 1, 3, ...
2, 4, 2, 4, 2, 4, 2, ...
1, 3, 1, 3, 1, 3, ...
2, 4, 2, 4, 2, ...
1, 3, 1, 3, ...
2, 4, 2, ...
1, 3, ...
2, ...
...
The sequence written as a triangle begins:
1;
2, 3;
1, 4, 1;
2, 3, 2, 3;
1, 4, 1, 4, 1;
2, 3, 2, 3, 2, 3;
1, 4, 1, 4, 1, 4, 1;
2, 3, 2, 3, 2, 3, 2, 3;
1, 4, 1, 4, 1, 4, 1, 4, 1;
2, 3, 2, 3, 2, 3, 2, 3, 2, 3;
...

Cf. A000034, A008851, A010684, A010685, A010693, A010694, A010702, A274912, A274921.

Table[1 + Boole@ EvenQ@ # + 2 Boole@ EvenQ@ k &[n - k + 1], {n, 14}, {k, n}] // Flatten (* Michael De Vlieger, Nov 14 2016 *)

a(n) = A274912(n) + 1.

A283393 a(n) = gcd(n^2-1, n^2+9).

Original entry on oeis.org

1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10, 1, 10, 1, 2, 5, 2, 5, 2, 1, 10

Offset: 0

Bruno Berselli, Mar 07 2017

Periodic with period 10.
Similar sequences with formula gcd(n^2-1, n^2+k):
k= 1:  1,  2, 1, 2, 1,  2, 1,  2, 1,  2, 1,  2, 1, ...  (A000034)
k= 3:  1,  4, 1, 4, 1,  4, 1,  4, 1,  4, 1,  4, 1, ...  (A010685)
k= 5:  1,  6, 3, 2, 3,  6, 1,  6, 3,  2, 3,  6, 1, ...  (A129203, start 6)
k= 7:  1,  8, 1, 8, 1,  8, 1,  8, 1,  8, 1,  8, 1, ...  (A010689)
k= 9:  1, 10, 1, 2, 5,  2, 5,  2, 1, 10, 1, 10, 1, ...  (this sequence)
k=11:  1, 12, 3, 4, 3, 12, 1, 12, 3,  4, 3, 12, 1, ...  (A129197, start 12)
Connection between the values of a(n) and the last digit of n:
. if n ends with 0, 2 or 8, then a(n) = 1;
. if n ends with 1 or 9, then a(n) = 10;
. if n ends with 3, 5 or 7, then a(n) = 2;
. if n ends with 4 or 6, then a(n) = 5.
Also, continued fraction expansion of (57 + sqrt(4579))/114.

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

Cf. A000034, A010685, A010689, A129197, A129203.

&cat [[1, 10, 1, 2, 5, 2, 5, 2, 1, 10]^^10];

Table[PolynomialGCD[n^2 - 1, n^2 + 9], {n, 0, 100}]
LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 10, 1, 2, 5, 2, 5, 2, 1, 10}, 100]

makelist(gcd(n^2-1, n^2+9), n, 0, 100);

Vec((1 + 10*x + x^2 + 2*x^3 + 5*x^4 + 2*x^5 + 5*x^6 + 2*x^7 + x^8 + 10*x^9)/(1 - x^10) + O(x^100)) \\ Colin Barker, Mar 08 2017

Python
```
[1, 10, 1, 2, 5, 2, 5, 2, 1, 10]*10
```

[gcd(n^2-1, n^2+9) for n in range(100)]

G.f.: (1 + 10*x + x^2 + 2*x^3 + 5*x^4 + 2*x^5 + 5*x^6 + 2*x^7 + x^8 + 10*x^9)/(1 - x^10).

A083589 Expansion of 1/((1-4x)(1-x^4)).

Original entry on oeis.org

1, 4, 16, 64, 257, 1028, 4112, 16448, 65793, 263172, 1052688, 4210752, 16843009, 67372036, 269488144, 1077952576, 4311810305, 17247241220, 68988964880, 275955859520, 1103823438081, 4415293752324, 17661175009296, 70644700037184

Offset: 0

Paul Barry, May 02 2003

G. C. Greubel, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (4,0,0,1,-4)

Cf. A033139, A000975.

CoefficientList[Series[1/((1-4x)(1-x^4)),{x,0,30}],x] (* or *) LinearRecurrence[ {4,0,0,1,-4},{1,4,16,64,257},31] (* Harvey P. Dale, Sep 13 2011 *)

a(n)=(4^(n+4)+64)\255 \\ Charles R Greathouse IV, Jul 09 2013

a(0)=1, a(n) = 4*a(n-1) if n is not a multiple of 4, otherwise a(n) = 4*a(n-1) + 1. - Vincenzo Librandi, Mar 19 2011
a(n) = 4^(n+4)/255 -1/12 +(-1)^n/20 +(-1)^floor(n/2)*A010685(n)/34. - R. J. Mathar, Mar 19 2011
a(0)=1, a(1)=4, a(2)=16, a(3)=64, a(4)=257, a(n) = 4*a(n-1) + a(n-4) - 4*a(n-5). - Harvey P. Dale, Sep 13 2011
a(n) = floor(64*(2^(2*(n+1))+1)/255). - Tani Akinari, Jul 09 2013

A166517 a(n) = (3 + 5(-1)^n + 6n)/4.

Original entry on oeis.org

2, 1, 5, 4, 8, 7, 11, 10, 14, 13, 17, 16, 20, 19, 23, 22, 26, 25, 29, 28, 32, 31, 35, 34, 38, 37, 41, 40, 44, 43, 47, 46, 50, 49, 53, 52, 56, 55, 59, 58, 62, 61, 65, 64, 68, 67, 71, 70, 74, 73, 77, 76, 80, 79, 83, 82, 86, 85, 89, 88, 92, 91, 95, 94, 98, 97, 101, 100, 104, 103, 107

Offset: 0

Vincenzo Librandi, Oct 16 2009

A sequence defined by a(1)=1, a(n)=k*n-a(n-1), k a constant parameter, has recurrence a(n)= 3*a(n-1) -3*a(n-2) +a(n-3). Its generating function is x*(1+2*(k-1)*x+(1-k)*x^2)/((1+x)*(1-x)^2). The closed form is a(n) = k*n/2+k/4+(-1)^n*(3*k/4-1). This applies with k=3 to this sequence here, and for example to sequences A165033, and A166519-A166525. - R. J. Mathar, Oct 17 2009
From Paul Curtz, Feb 20 2010: (Start)
Also: A001651, terms swapped by pairs.
a(n) mod 9 defines a period-6 sequence which is a permutation of A141425. (End)

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A001651, A010685, A010716, A016777, A016945, A073010.

[(3 +5*(-1)^n+6*n)/4: n in [0..80]]; // Vincenzo Librandi, Sep 13 2013

CoefficientList[Series[(2 x^2 - x + 2)/((1 + x) (x - 1)^2), {x, 0, 80}], x] (* Harvey P. Dale, Mar 25 2011 *)
Table[(3 + 5 (-1)^n + 6 n) / 4, {n, 0, 100}] (* Vincenzo Librandi, Sep 13 2013 *)

a(n) = 3*n - a(n-1).
From Paul Curtz, Feb 20 2010: (Start)
a(n+1)-a(n) = (-1)^(n+1)*A010685(n).
Second differences: |a(n+2)-2*a(n+1)+a(n)| = A010716(n).
a(2*n) + a(2*n+1) = A016945(n) = 6*n+3.
a(2*n) = A016945(n).
a(2*n+1) = A016777(n). (End)
G.f. ( 2-x+2*x^2 ) / ( (1+x)*(x-1)^2 ). - R. J. Mathar, Mar 08 2011
E.g.f.: (1/4)*exp(-x)*(5 + 3*exp(2*x) + 6*x*exp(2*x)). - G. C. Greubel, May 15 2016
Sum_{n>=0} (-1)^(n+1)/a(n) = Pi/(3*sqrt(3)) (A073010). - Amiram Eldar, Feb 24 2023

a(0)=2 added by Paul Curtz, Feb 20 2010

A140359 a(n) = 2a(n-1) + a(n-2) - 2a(n-3).

Original entry on oeis.org

1, 1, 6, 11, 26, 51, 106, 211, 426, 851, 1706, 3411, 6826, 13651, 27306, 54611, 109226, 218451, 436906, 873811, 1747626, 3495251, 6990506, 13981011, 27962026, 55924051, 111848106, 223696211, 447392426, 894784851, 1789569706, 3579139411

Offset: 0

Paul Curtz, Jun 24 2008

nonn

This is the sequence A(1,1;1,2;3) of the family of sequences [a,b:c,d:k] considered by G. Detlefs, and treated as A(a,b;c,d;k) in the W. Lang link given below. - Wolfdieter Lang, Oct 18 2010

G. C. Greubel, Table of n, a(n) for n = 0..1000
Wolfdieter Lang, Notes on certain inhomogeneous three term recurrences.
Index entries for linear recurrences with constant coefficients, signature (2,1,-2).

Cf. A000975, A001045, A087288, A137241, A140360.

[(5*2^(n+1) -9 + 5*(-1)^n)/6: n in [0..50]]; // G. C. Greubel, Oct 10 2017

Table[(5*2^(n+1) -9 + 5*(-1)^n)/6, {n, 0, 50}] (* G. C. Greubel, Oct 10 2017 *)
LinearRecurrence[{2,1,-2},{1,1,6},40] (* Harvey P. Dale, Mar 24 2021 *)

for(n=0,50, print1((5*2^(n+1) -9 + 5*(-1)^n)/6, ", ")) \\ G. C. Greubel, Oct 10 2017

a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
a(n+1) - a(n) = 5*A001045(n), Jacobsthal numbers.
a(n+1) - 2*a(n) = (-1)^(n+1)* A010685(n).
From R. J. Mathar, Jul 10 2008: (Start)
O.g.f.: (1-x+3*x^2)/((x-1)*(2*x-1)*(1+x)).
a(n) = (5*2^(n+1) - 9 + 5*(-1)^n)/6. (End)
a(n) = a(n-1) + 2*a(n-2) +3, n>1 - Gary Detlefs, Jun 20 2010

Extended by R. J. Mathar, Jul 10 2008

A171478 a(n) = 6a(n-1) - 8a(n-2) + 2 for n > 1; a(0) = 1, a(1) = 8.

Original entry on oeis.org

1, 8, 42, 190, 806, 3318, 13462, 54230, 217686, 872278, 3492182, 13974870, 55911766, 223671638, 894735702, 3579041110, 14316361046, 57265837398, 229064136022, 916258116950, 3665035613526, 14660148745558, 58640607565142

Offset: 0

Klaus Brockhaus, Dec 09 2009

Second binomial transform of A168648.

Partial sums of A080960.

Harvey P. Dale, Table of n, a(n) for n = 0..1000 (extending prior file from Vincenzo Librandi)
Index entries for linear recurrences with constant coefficients, signature (7,-14,8).

Cf. A168648 ((10*2^n+2*(-1)^n)/3, a(0)=1), A080960 (third binomial transform of A010685), A171472, A171473.

a:=[1,8];; for n in [3..25] do a[n]:=6*a[n-1]-8*a[n-2]+2; od; a; # Muniru A Asiru, Mar 22 2018

[(10*4^n-9*2^n+2)/3: n in [0..30]]; // Vincenzo Librandi, Jul 18 2011

a:= proc(n) option remember: if n = 0 then 1 elif n = 1 then 8 elif  n >= 2 then 6*procname(n-1) - 8*procname(n-2) + 2 fi; end:
seq(a(n), n = 0..25); # Muniru A Asiru, Mar 22 2018

RecurrenceTable[{a[0]==1,a[1]==8,a[n]==6a[n-1]-8a[n-2]+2},a,{n,30}] (* or *) LinearRecurrence[{7,-14,8},{1,8,42},30] (* Harvey P. Dale, May 04 2012 *)

{m=23; v=concat([1, 8], vector(m-2)); for(n=3, m, v[n]=6*v[n-1]-8*v[n-2]+2); v}

a(n) = (10*4^n - 9*2^n + 2)/3.
G.f.: (1+x)/((1-x)*(1-2*x)*(1-4*x)).
a(0)=1, a(1)=8, a(2)=42, a(n) = 7*a(n-1) - 14*a(n-2) + 8*a(n-3). - Harvey P. Dale, May 04 2012
a(n) = A203241(n+1) + 2^n*(2^(n+1)-1), n>0. - J. M. Bergot, Mar 21 2018

A174571 a(4n)=n, a(4n+1)=4, a(4n+2)=1, a(4n+3)=4.

Original entry on oeis.org

0, 4, 1, 4, 1, 4, 1, 4, 2, 4, 1, 4, 3, 4, 1, 4, 4, 4, 1, 4, 5, 4, 1, 4, 6, 4, 1, 4, 7, 4, 1, 4, 8, 4, 1, 4, 9, 4, 1, 4, 10, 4, 1, 4, 11, 4, 1, 4, 12, 4, 1, 4, 13, 4, 1, 4, 14, 4, 1, 4, 15, 4, 1, 4, 16, 4, 1, 4, 17, 4, 1, 4, 18, 4, 1, 4, 19, 4, 1, 4, 20, 4, 1, 4

Offset: 0

Paul Curtz, Nov 29 2010

Antti Karttunen, Table of n, a(n) for n = 0..16383
Index entries for linear recurrences with constant coefficients, signature (0,0,0,2,0,0,0,-1).

Cf. A010685.

[(n mod 4) eq 0 select n/4 else Modexp(4,n,5): n in [0..90]]; // G. C. Greubel, Nov 23 2021

Array[Which[OddQ@ Mod[#, 4], 4, Mod[#, 4] == 0, #/4, True, 1] &, 84, 0] (* or *)
CoefficientList[Series[x*(4 +x +4*x^2 +x^3 -4*x^4 -x^5 -4*x^6)/(1-x^4)^2, {x, 0, 83}], x] (* Michael De Vlieger, Nov 06 2018 *)
LinearRecurrence[{0,0,0,2,0,0,0,-1},{0,4,1,4,1,4,1,4},100] (* Harvey P. Dale, Dec 21 2024 *)

A174571(n) = if(!(n%4),n/4,if(2==(n%4),1,4)); \\ Antti Karttunen, Nov 06 2018

def A174571(n): return n/4 if (n%4==0) else power_mod(4,n,5)
[A174571(n) for n in (0..90)] # G. C. Greubel, Nov 23 2021

a(n) = A010685(n) if 4 does not divide n.
a(n) = 2*a(n-4) - a(n-8).
G.f.: x*(4 + x + 4*x^2 + x^3 - 4*x^4 - x^5 - 4*x^6)/( (1-x)*(1+x)*(1+x^2) )^2.
a(n) = (36 +n +(n-28)*(-1)^n +2*(n -5 +(-1)^n)*cos(n*Pi/2) +(1+(-1)^n)*sin(n*Pi/2) )/16. - Wesley Ivan Hurt, May 07 2021
E.g.f.: (1/8)*(4*cosh(x) + (x+32)*sinh(x) - 4*cos(x) - x*sin(x)). - G. C. Greubel, Nov 23 2021

cp's OEIS Frontend

A040005 Continued fraction for sqrt(8).

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A094015 Expansion of (1+4*x)/(1-8*x^2).

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A010698 Period 2: repeat (2,8).

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A274913 Square array read by antidiagonals upwards in which each new term is the least positive integer distinct from its neighbors.

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A283393 a(n) = gcd(n^2-1, n^2+9).

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A083589 Expansion of 1/((1-4*x)*(1-x^4)).

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A166517 a(n) = (3 + 5*(-1)^n + 6*n)/4.

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A094015 Expansion of (1+4x)/(1-8x^2).

A083589 Expansion of 1/((1-4x)(1-x^4)).

A166517 a(n) = (3 + 5(-1)^n + 6n)/4.

A140359 a(n) = 2a(n-1) + a(n-2) - 2a(n-3).

A171478 a(n) = 6a(n-1) - 8a(n-2) + 2 for n > 1; a(0) = 1, a(1) = 8.