A107453 -id:A107453 - cp's OEIS frontend

A177704 Period 4: repeat [1, 1, 1, 2].

1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1

Offset: 0

Klaus Brockhaus, May 11 2010

Continued fraction expansion of (3 + 2*sqrt(6))/5.
Decimal expansion of 1112/9999.
a(n) = A164115(n + 1) = (-1)^(n + 1) * A164117(n + 1) = A138191(n + 3) = A107453(n + 5).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
David Nacin, Van der Laan Sequences and a Conjecture on Padovan Numbers, J. Int. Seq., Vol. 26 (2023), Article 23.1.2.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A001068, A107453, A138191, A164115, A164117, A177705, A219977, A236398.

Magma
```
&cat[ [1, 1, 1, 2]: k in [1..27] ];
```

A177704:=n->floor((n+1)*5/4) - floor(n*5/4): seq(A177704(n), n=0..100); # Wesley Ivan Hurt, Jun 15 2016

Table[Floor[(n + 1)*5/4] - Floor[n*5/4], {n, 0, 100}] (* Wesley Ivan Hurt, Jun 15 2016 *)
LinearRecurrence[{0, 0, 0, 1}, {1, 1, 1, 2}, 100] (* Vincenzo Librandi, Jun 16 2016 *)

a(n) = if(n%4==3, 2, 1) \\ Felix Fröhlich, Jun 15 2016

a(n) = (5-(-1)^n + i*i^n-i*(-i)^n)/4 where i = sqrt(-1).
a(n) = a(n-4) for n > 3; a(0) = 1, a(1) = 1, a(2) = 1, a(3) = 2.
G.f.: (1+x+x^2+2*x^3)/(1-x^4).
a(n) = 1 + (1-(-1)^n) * (1+i^(n+1))/4 where i = sqrt(-1). - Bruno Berselli, Apr 05 2011
a(n) = 5/4 - sin(Pi*n/2)/2 - (-1)^n/4. - R. J. Mathar, Oct 08 2011
a(n) = floor((n+1)*5/4) - floor(n*5/4). - Hailey R. Olafson, Jul 23 2014
From Wesley Ivan Hurt, Jun 15 2016: (Start)
a(n+3) - a(n+2) = A219977(n).
Sum_{i=0..n-1} a(i) = A001068(n). (End)
E.g.f.: (-sin(x) + 3*sinh(x) + 2*cosh(x))/2. - Ilya Gutkovskiy, Jun 15 2016

A107452 Number of nonisomorphic bipartite generalized Petersen graphs P(2n,k) on 4n vertices for 1<=k

Original entry on oeis.org

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

Offset: 2

Marko Boben (Marko.Boben(AT)fmf.uni-lj.si), Tomaz Pisanski and Arjana Zitnik (Arjana.Zitnik(AT)fmf.uni-lj.si), May 26 2005

nonn

The generalized Petersen graph P(n,k) is a graph with vertex set V(P(n,k)) = {u_0,u_1,...,u_{n-1},v_0,v_1,...,v_{n-1}} and edge set E(P(n,k)) = {u_i u_{i+1}, u_i v_i, v_i v_{i+k} : i=0,...,n-1}, where the subscripts are to be read modulo n.

			A generalized Petersen graph P(n,k) is bipartite if and only if n is even and k is odd.
The smallest bipartite generalized Petersen graph is P(4,1)

I. Z. Bouwer, W. W. Chernoff, B. Monson and Z. Star, The Foster Census (Charles Babbage Research Centre, 1988), ISBN 0-919611-19-2.

Marko Boben, Tomaz Pisanski and Arjana Zitnik, I-graphs and the corresponding configurations, Preprint series (University of Ljubljana, IMFM), Vol. 42 (2004), 939 (ISSN 1318-4865).
B. Horvat, T. Pisanski; A. Zitnik. Isomorphism checking of I-graphs, Graphs Comb. 28, No. 6, 823-830 (2012).
M. Watkins, A theorem on Tait colorings with an application to the generalized Petersen graphs, J. Combin. Theory 6 (1969), 152-164.

Cf. A077105, A107453-A107460.

A138191 Denominator of (n-1)n(n+1)/12.

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1

Offset: 1

Eric W. Weisstein, Mar 04 2008

Proof of 4-periodicity follows from evaluating (n+3)(n+4)(n+5)/12, subtracting (n-1)n(n+1)/12 and getting n^2+4n+5 which is an integer. - R. J. Mathar, Mar 07 2008

			0, 1/2, 2, 5, 10, 35/2, 28, 42, 60, 165/2, 110, 143, 182, ...

Eric Weisstein's World of Mathematics, Kirchhoff Index.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A000292, A107453, A107453, A138190.

Table[(n^3-n)/12,{n,120}]//Denominator (* or *) PadRight[{},120,{1,2,1,1}] (* Harvey P. Dale, Apr 15 2019 *)

def A138191(n): return (1,1,2,1)[n&3] # Chai Wah Wu, Apr 25 2024

From R. J. Mathar, Mar 07 2008: (Start)
a(n) = 1 + (A000292(n-1) mod 2) = a(n-4).
O.g.f.: -1-5/(4(x-1))+1/(4(x+1))-1/(2(x^2+1)). (End)
From Amiram Eldar, Jan 01 2023: (Start)
Multiplicative with a(p^e) = 2 if p = 2 and e = 1, and 1 otherwise.
Dirichlet g.f.: zeta(s)*(1+1/2^s-1/4^s).
Sum_{k=1..n} a(k) ~ (5/4)*n. (End)

A164115 Expansion of (1 - x^5) / ((1 - x) * (1 - x^4)) in powers of x.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2

Offset: 0

Michael Somos, Aug 10 2009

The sequence A107453 has the same terms but different offset.
Convolution inverse of A164116.
Decimal expansion of 11111/99990. - Elmo R. Oliveira, Feb 18 2024

			1 + x + x^2 + x^3 + 2*x^4 + x^5 + x^6 + x^7 + 2*x^8 + x^9 + x^10 + ...

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

Cf. A107453, A138191, A164116, A164117.

m:=100; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1+x+x^2+x^3+x^4)/(1-x^4))); // G. C. Greubel, Sep 22 2018

CoefficientList[Series[(1+x+x^2+x^3+x^4)/(1-x^4), {x, 0, 100}], x] (* G. C. Greubel, Sep 22 2018 *)
LinearRecurrence[{0,0,0,1},{1,1,1,1,2},120] (* or *) PadRight[{1},120,{2,1,1,1}] (* Harvey P. Dale, Aug 24 2019 *)

PARI
```
{a(n) = 2 - (n==0) - (n%4>0)}
```

x='x+O('x^99); Vec((1-x^5)/((1-x)*(1-x^4))) \\ Altug Alkan, Sep 23 2018

Euler transform of length-5 sequence [ 1, 0, 0, 1, -1].
a(n) is multiplicative with a(2) = 1, a(2^e) = 2 if e>1, a(p^e) = 1 if p>2.
a(n) = (-1)^n * A164117(n).
a(4*n) = 2 unless n=0. a(2*n + 1) = a(4*n + 2) = 1.
a(-n) = a(n). a(n+4) = a(n) unless n=0 or n=-4.
G.f.: (1 + x + x^2 + x^3 + x^4) / ((1+x)*(1-x)*(1+x^2)).
a(n) = A138191(n+2), n>0. - R. J. Mathar, Aug 17 2009
Dirichlet g.f. (1+1/4^s)*zeta(s). - R. J. Mathar, Feb 24 2011
a(n) = (i^n + (-i)^n + (-1)^n + 5)/4 for n > 0 where i is the imaginary unit. - Bruno Berselli, Feb 25 2011

cp's OEIS Frontend

A177704 Period 4: repeat [1, 1, 1, 2].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A107452 Number of nonisomorphic bipartite generalized Petersen graphs P(2n,k) on 4n vertices for 1<=k

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

A138191 Denominator of (n-1)*n*(n+1)/12.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Formula

A164115 Expansion of (1 - x^5) / ((1 - x) * (1 - x^4)) in powers of x.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

A138191 Denominator of (n-1)n(n+1)/12.