A090458 -id:A090458 - cp's OEIS frontend

A176040 Periodic sequence: Repeat 3, 1.

3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3

Offset: 0

Klaus Brockhaus, Apr 07 2010

Interleaving of A010701 and A000012.
Also continued fraction expansion of (3+sqrt(21))/2.
Also decimal expansion of 31/99.
Essentially first differences of A014601.
Inverse binomial transform of 3 followed by A020707.
Second inverse binomial transform of A052919 without initial term 2.
Third inverse binomial transform of A007582 without initial term 1.
Exp( Sum_{n >= 1} a(n)*x^n/n ) = 1 + x + 2*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + ... is the o.g.f. for A008619. - Peter Bala, Mar 13 2015

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

Cf. A153284, A010701 (all 3's sequence), A000012 (all 1's sequence), A090458 (decimal expansion of (3+sqrt(21))/2), A010684 (repeat 1, 3), A014601 (congruent to 0 or 3 mod 4), A020707 (2^(n+2)), A052919, A007582 (2^(n-1)*(1+2^n)), A008619.

&cat[ [3, 1]: n in [0..52] ];
[ 2+(-1)^n: n in [0..104] ];

PadRight[{},120,{3,1}] (* or *) LinearRecurrence[{0,1},{3,1},120] (* Harvey P. Dale, Mar 11 2015 *)

a(n) = 2+(-1)^n.
a(n) = a(n-2) for n > 1; a(0) = 3, a(1) = 1.
a(n) = -a(n-1)+4 for n > 0; a(0) = 3.
a(n) = 3*((n+1) mod 2)+(n mod 2).
a(n) = A010684(n+1).
G.f.: (3+x)/((1-x)*(1+x)).
From Amiram Eldar, Jan 01 2023: (Start)
Multiplicative with a(2^e) = 3, and a(p^e) = 1 for p >= 3.
Dirichlet g.f.: zeta(s)*(1+2^(1-s)). (End)

A222135 Decimal expansion of sqrt(5 - sqrt(5 - sqrt(5 - sqrt(5 - ... )))).

Original entry on oeis.org

1, 7, 9, 1, 2, 8, 7, 8, 4, 7, 4, 7, 7, 9, 2, 0, 0, 0, 3, 2, 9, 4, 0, 2, 3, 5, 9, 6, 8, 6, 4, 0, 0, 4, 2, 4, 4, 4, 9, 2, 2, 2, 8, 2, 8, 8, 3, 8, 3, 9, 8, 5, 9, 5, 1, 3, 0, 3, 6, 2, 1, 0, 6, 1, 9, 5, 3, 4, 3, 4, 2, 1, 2, 7, 7, 3, 8, 8, 5, 4, 4, 3, 3, 0, 2, 1, 8, 0, 7, 7, 9, 7, 4, 6, 7, 2, 2, 5, 1, 6, 3

Offset: 1

Jaroslav Krizek, Feb 08 2013

Sequence with a(1) = 2 is decimal expansion of sqrt(5 + sqrt(5 + sqrt(5 + sqrt(5 + ... )))) - A222134.

			1.791287847477920003294023596864...

Cf. A090458, A107905, A222134.

Mathematica
```
RealDigits[(Sqrt[21] - 1)/2, 10, 130]
```

Closed form: (sqrt(21) - 1)/2 = A090458-2 = A107905-3 = A222134-1.
sqrt(5 - sqrt(5 - sqrt(5 - sqrt(5 - ... )))) + 1 = sqrt(5 + sqrt(5 + sqrt(5 + sqrt(5 + ... )))). See A222134.
Minimal polynomial: x^2 + x - 5. - Stefano Spezia, Jul 02 2025

A134927 a(0)=a(1)=1; a(n) = 3*(a(n-1) + a(n-2)).

Original entry on oeis.org

1, 1, 6, 21, 81, 306, 1161, 4401, 16686, 63261, 239841, 909306, 3447441, 13070241, 49553046, 187869861, 712268721, 2700415746, 10238053401, 38815407441, 147160382526, 557927369901, 2115263257281, 8019571881546, 30404505416481

Offset: 0

Rolf Pleisch, Jan 29 2008

Joshua Zucker, Feb 23 2008, Table of n, a(n) for n = 0..50
Index entries for linear recurrences with constant coefficients, signature (3,3).

Essentially the same as A108306.

a[0]:=1:a[1]:=1:for n from 2 to 50 do a[n]:=3*a[n-1]+3*a[n-2] od: seq(a[n], n=0..33); # Zerinvary Lajos, Dec 14 2008

Mathematica
```
LinearRecurrence[{3, 3}, {1, 1}, 30]
```

a=[1,1];for(i=2,10,a=concat(a,3*a[#a]+3*a[#a-1]));a \\ Charles R Greathouse IV, Oct 04 2011

from sage.combinat.sloane_functions import recur_gen2
it = recur_gen2(1,1,3,3)
[next(it) for i in range(25)] # Zerinvary Lajos, Jun 25 2008

From R. J. Mathar, Jan 31 2008: (Start)
O.g.f.: (-1+2*x)/(-1 + 3*x + 3*x^2).
a(n) = A030195(n+1)-2*A030195(n). (End)
a(n) = A108306(n-1), n>0. - R. J. Mathar, Oct 04 2011
a(n) ~ 3.7912878474...^n, where the constant is A090458. - Charles R Greathouse IV, Oct 04 2011

More terms from Joshua Zucker, Feb 23 2008

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A176040 Periodic sequence: Repeat 3, 1.

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A222135 Decimal expansion of sqrt(5 - sqrt(5 - sqrt(5 - sqrt(5 - ... )))).

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A134927 a(0)=a(1)=1; a(n) = 3*(a(n-1) + a(n-2)).

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