A005009 -id:A005009 - cp's OEIS frontend

A176355 Periodic sequence: Repeat 6, 1.

6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6

Offset: 0

Klaus Brockhaus, Apr 15 2010

Interleaving of A010722 and A000012.
Also continued fraction expansion of 3+sqrt(15).
Also decimal expansion of 61/99.
Essentially first differences of A047335.
Binomial transform of 6 followed by A166577 without initial terms 1, 4.
Inverse binomial transform of A005009 preceded by 6.

			0.6161616161616161616161616161616161616161...

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

Cf. A010722 (all 6's sequence), A000012 (all 1's sequence), A092294 (decimal expansion of 3+sqrt(15)), A010687 (repeat 1, 6), A047335 (congruent to 0 or 6 mod 7), A166577, A005009 (7*2^n).

Magma
```
&cat[ [6, 1]: n in [0..52] ];
```
Magma
```
[(7+5*(-1)^n)/2: n in [0..104]];
```

PadRight[{},120,{6,1}] (* Harvey P. Dale, Apr 12 2018 *)

G.f.: (6 + x)/(1 - x^2).
a(n) = (7 + 5*(-1)^n)/2.
a(n) = a(n-2) for n>1, a(0)=6, a(1)=1.
a(n) = -a(n-1)+7 for n>0, a(0)=6.
a(n) = 6*((n+1) mod 2) + (n mod 2).
a(n) = A010687(n+1).
a(n) = 13^n mod 7. - Vincenzo Librandi, Jun 01 2016
From Amiram Eldar, Jan 01 2023: (Start)
Multiplicative with a(2^e) = 6, and a(p^e) = 1 for p >= 3.
Dirichlet g.f.: zeta(s)*(1+5/2^s). (End)
E.g.f.: 6*cosh(x) + sinh(x). - Stefano Spezia, Feb 09 2025

A204543 Number of (n+1)X2 0..1 arrays with permanents of 2X2 subblocks differing from neighboring permanents.

Original entry on oeis.org

16, 26, 41, 56, 80, 112, 162, 224, 322, 448, 646, 896, 1290, 1792, 2582, 3584, 5162, 7168, 10326, 14336, 20650, 28672, 41302, 57344, 82602, 114688, 165206, 229376, 330410, 458752, 660822, 917504, 1321642, 1835008, 2643286, 3670016, 5286570

Offset: 1

R. H. Hardin Jan 16 2012

nonn

			Some solutions for n=8
..1..1....1..1....1..1....0..0....0..1....0..0....1..1....1..0....0..1....1..0
..1..0....0..1....1..0....0..1....1..0....1..1....1..0....1..0....1..0....0..1
..1..0....0..1....1..0....1..1....1..0....0..1....1..0....1..1....1..0....0..1
..1..1....1..1....0..1....1..1....0..1....0..1....0..1....1..1....1..1....1..0
..1..1....1..1....0..1....1..0....0..1....1..1....0..1....1..0....1..1....1..0
..0..1....0..1....1..1....1..0....1..1....1..1....1..1....1..0....0..1....0..1
..0..1....0..1....1..1....0..1....1..1....0..1....1..1....1..1....0..1....0..1
..1..0....1..0....0..1....0..1....0..1....0..1....1..0....1..1....1..1....1..1
..1..0....0..0....0..1....1..0....0..0....1..1....0..0....1..0....0..0....0..0

R. H. Hardin, Table of n, a(n) for n = 1..210

Even terms are A005009(n/2+1) for n>=4

Empirical: a(n) = a(n-2) +2*a(n-4) for n>6

Empirical: a(2k)=14*2^k for k>1

A213255 2^(n-1) - floor((2^(n-1) - 1)/(n-1)).

Original entry on oeis.org

1, 3, 6, 13, 26, 54, 110, 225, 456, 922, 1862, 3755, 7562, 15214, 30584, 61441, 123362, 247581, 496694, 996148, 1997288, 4003654, 8023886, 16078166, 32212255, 64527754, 129246702, 258848476, 518358122, 1037950430, 2078209982, 4160749569, 8329633544

Offset: 2

Arkadiusz Wesolowski, Jun 07 2012

Lower bounds of the decycling numbers of n-cubes for n >= 9.

			a(8) = 110 because 2^7 - (2^7 - 1)/7 = 109.8571428571....

Arkadiusz Wesolowski, Table of n, a(n) for n = 2..800
Sheng Bau, The Decycling Number of Graphs

Cf. A005009.

[Ceiling(2^(n-1)-(2^(n-1)-1)/(n-1)) : n in [2..34]];

Table[Ceiling[2^(n - 1) - (2^(n - 1) - 1)/(n - 1)], {n, 2, 34}]

for(n=2, 34, print1(ceil(2^(n-1)-(2^(n-1)-1)/(n-1)), ", "))

a(n) = 2^(n-1) - floor((2^(n-1) - 1)/(n-1)).

a(n) = ceiling(2^(n-1) - (2^(n-1) - 1)/(n-1)).

A275788 a(0) = 0, a(n+1) = 2*a(n) + (-1)^floor(n/3).

Original entry on oeis.org

0, 1, 3, 7, 13, 25, 49, 99, 199, 399, 797, 1593, 3185, 6371, 12743, 25487, 50973, 101945, 203889, 407779, 815559, 1631119, 3262237, 6524473, 13048945, 26097891, 52195783, 104391567, 208783133, 417566265, 835132529, 1670265059, 3340530119, 6681060239

Offset: 0

Paul Curtz, Aug 09 2016

nonn

a(n) and its successive differences:
0,  1,  3,  7, 13, 25, 49, ...
1,  2,  4,  6, 12, 24, 50, 100, ...
1,  2,  2,  6, 12, 26, 50, 100, 198, ...
1,  0,  4,  6, 14, 24, 50,  98, 200, 398, ...
-1, 4,  2,  8, 10, 26, 48, 102, 198, 400, 794, ...
5, -2,  6,  2, 16, 22, 54,  96, 202, 394, 800, 1590, ...
-7, 8, -4, 14,  6, 32, 42, 106, 192, 406, 790, 1600, 3178, ...
... .
Each row has the recurrence a(n) + a(n+3) = 7*2^n.
Main diagonal: 2*A001045(n).
Upper diagonals: A084214(n+1), 3*2^n, ... .
Subdiagonals: 2^n, A078008(n), A084214(n+1), -2^n, ... .
a(-n) = 0, 1/2, 3/4, 7/8, -1/16, -17/32, -49/64, 15/128, ... .
b(n), numerators of a(-n), and first differences:
0, 1, 3,  7,  -1, -17, -49,  15, 143,  399,  -113, -1137, ...
1, 2, 4, -8, -16, -32,  64, 128, 256, -512, -1024, ... = A000079(n)*A130151(n), not in the OEIS.

			a(1)=2*0+1=1, a(2)=2*1+1=3, a(2)=2*3+1=7, a(3)=2*7-1=13, a(4)=2*13-1=25, ... .

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,-1,2).

Cf. A000079, A001045, A002264, A005009, A007283, A078008, A084214, A113405, A130151, A274817.

CoefficientList[Series[x (1 + x + x^2)/((1 + x) (1 - 2 x) (1 - x + x^2)), {x, 0, 33}], x] (* Michael De Vlieger, Aug 11 2016 *)
LinearRecurrence[{2,0,-1,2}, {0, 1, 3, 7}, 25] (* G. C. Greubel, Aug 16 2016 *)

concat(0, Vec(x*(1+x+x^2)/((1+x)*(1-2*x)*(1-x+x^2)) + O(x^40))) \\ Colin Barker, Aug 10 2016

From Colin Barker, Aug 09 2016: (Start)
a(n) = 2*a(n-1) - a(n-3) + 2*a(n-4) for n>3.
G.f.: x*(1 + x + x^2) / ((1+x)*(1-2*x)*(1-x+x^2)).
(End)
a(n+3) = 7*2^n - a(n), a(0)=0, a(1)=1, a(2)=3.

More terms from Colin Barker, Aug 10 2016

A365062 Enumeration of | Sort_n(123,321) |.

Original entry on oeis.org

1, 1, 2, 4, 7, 14, 28, 56, 112, 224, 448, 896, 1792, 3584, 7168, 14336, 28672, 57344, 114688, 229376, 458752, 917504, 1835008, 3670016, 7340032, 14680064, 29360128, 58720256, 117440512, 234881024, 469762048, 939524096, 1879048192, 3758096384, 7516192768, 15032385536

Offset: 0

Michael De Vlieger, Aug 23 2023

Christopher Bao, Yunseo Choi, Katelyn Gan, and Owen Zhang, On a Conjecture by Baril, Cerbai, Khalil, and Vajnovszki on Two Restricted Stacks, arXiv:2308.09344 [math.CO], 2023. See Theorems 1.1 and 1.2, pp. 2-3.
Index entries for linear recurrences with constant coefficients, signature (2).

Cf. A000079, A005009.

{1}~Join~Array[If[# <= 3, 2^(# - 1), 7*2^(# - 4)] &, 30]

a(0) = 1, a(n) = 2^(n-1) for n = 1..3, a(n) = 7*2^(n-4) for n > 3.
From Chai Wah Wu, Aug 24 2023: (Start)
a(n) = 2*a(n-1) for n > 4.
G.f.: (x^4 + x - 1)/(2*x - 1). (End)

A220753 Expansion of (1+4x+5x^2-x^3)/((1-x)(1+x)(1-2*x^2)).

Original entry on oeis.org

1, 4, 8, 11, 22, 25, 50, 53, 106, 109, 218, 221, 442, 445, 890, 893, 1786, 1789, 3578, 3581, 7162, 7165, 14330, 14333, 28666, 28669, 57338, 57341, 114682, 114685, 229370, 229373, 458746, 458749, 917498, 917501, 1835002, 1835005, 3670010, 3670013

Offset: 0

Philippe Deléham, Apr 13 2013

Index entries for linear recurrences with constant coefficients, signature (0,3,0,-2).

Cf. A005009, A048489, A063757.

m:=41; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+4*x+5*x^2-x^3)/((1-x)*(1+x)*(1-2*x^2)))); // Bruno Berselli, Apr 13 2013

Table[7 2^Floor[n/2] - (3/2) (3 + (-1)^n), {n, 0, 40}] (* Bruno Berselli, Apr 13 2013 *)
LinearRecurrence[{0, 3, 0, -2}, {1, 4, 8, 11}, 40] (* T. D. Noe, Apr 17 2013 *)

G.f.: (1+4*x+5*x^2-x^3)/((1-x)*(1+x)*(1-2*x^2)).
a(2n) = 7*2^n - 6 = A048489(n) = A063757(2n) = A005009(n)-6.
a(2n+1) = 7*2^n - 3 = A048489(n) + 3 = A063757(2n+1) - 3*A000225(n) = A005009(n)-3.
a(n) = a(n-1)*2 if n even.
a(n) = a(n-1)+3 if n odd.
a(n) = 3*a(n-2) - 2*a(n-4) with a(0)=1, a(1)=4, a(2)=8, a(3)=11.
a(n) = 7*2^floor(n/2) - (3/2)*(3+(-1)^n).
a(n) = A047290(A083416(n+1)). [Bruno Berselli, Apr 13 2013]

cp's OEIS Frontend

A176355 Periodic sequence: Repeat 6, 1.

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A204543 Number of (n+1)X2 0..1 arrays with permanents of 2X2 subblocks differing from neighboring permanents.

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A213255 2^(n-1) - floor((2^(n-1) - 1)/(n-1)).

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A275788 a(0) = 0, a(n+1) = 2*a(n) + (-1)^floor(n/3).

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A365062 Enumeration of | Sort_n(123,321) |.

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A220753 Expansion of (1+4*x+5*x^2-x^3)/((1-x)*(1+x)*(1-2*x^2)).

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Magma

Mathematica

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A220753 Expansion of (1+4x+5x^2-x^3)/((1-x)(1+x)(1-2*x^2)).