login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A028242 Follow n+1 by n. Also (essentially) Molien series of 2-dimensional quaternion group Q_8. 42
1, 0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, 8, 7, 9, 8, 10, 9, 11, 10, 12, 11, 13, 12, 14, 13, 15, 14, 16, 15, 17, 16, 18, 17, 19, 18, 20, 19, 21, 20, 22, 21, 23, 22, 24, 23, 25, 24, 26, 25, 27, 26, 28, 27, 29, 28, 30, 29, 31, 30, 32, 31, 33, 32, 34, 33, 35, 34, 36, 35, 37, 36, 38 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A two-way infinite sequences which is palindromic (up to sign). - Michael Somos, Mar 21, 2003

Number of permutations of [n+1] avoiding the patterns 123, 132 and 231 and having exactly one fixed point. Example: a(0) because we have 1; a(2)=2 because we have 213 and 321; a(3)=1 because we have 3214. - Emeric Deutsch, Nov 17 2005

The ring of invariants for the standard action of Quaternions on C^2 is generated by x^4 + y^4, x^2 * y^2, and x * y * (x^4 - y^4). - Michael Somos, Mar 14 2011

A000027 and A001477 interleaved. - Omar E. Pol, Feb 06 2012

First differences are A168361, extended by an initial -1. (Or: a(n)-a(n-1) = A168361(n+1), for all n >= 1.) - M. F. Hasler, Oct 05 2017

REFERENCES

D. Benson, Polynomial Invariants of Finite Groups, Cambridge, p. 23.

S. Mukai, An Introduction to Invariants and Moduli, Cambridge, 2003; see p. 15.

M. D. Neusel and L. Smith, Invariant Theory of Finite Groups, Amer. Math. Soc., 2002; see p. 97.

L. Smith, Polynomial Invariants of Finite Groups, A K Peters, 1995, p. 90.

LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 0..1000

H. W. Gould, The inverse of a finite series and a third-order recurrent sequence, Fibonacci Quart. 44 (2006), no. 4, 302-315. See page 311.

T. Mansour and A. Robertson, Refined restricted permutations avoiding subsets of patterns of length three, Annals of Combinatorics, 6, 2002, 407-418 (Theorem 3.3).

MathOverflow, A question about an application of Molien's formula to find the generators and relations of an invariant ring

Index entries for two-way infinite sequences

Index entries for Molien series

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

FORMULA

Expansion of the Molien series for standard action of Quaternions on C^2: (1 + t^6) / (1 - t^4)^2 = (1 - t^12) / ((1 - t^4)^2 * (1 - t^6)) in powers of t^2.

Euler transform of length 6 sequence [ 0, 2, 1, 0, 0, -1]. - Michael Somos, Mar 14 2011

a(n) = n-a(n-1) [with a(0) = 1] = A000035(n-1)+A004526(n). - Henry Bottomley, Jul 25 2001

G.f.: (1 - x + x^2) / ((1 - x) * (1 - x^2)) = ( 1+x^2-x ) / ( (1+x)*(x-1)^2 ). a(2*n) = n + 1, a(2*n + 1) = n, a(-1 - n) = -a(n). a(n) = a(n - 1) + a(n - 2) - a(n - 3).

a(n) = floor(n/2) + 1 - n mod 2. a(2*k) = k+1, a(2*k+1) = k; A110657(n) = a(a(n)), A110658(n) = a(a(a(n))); a(n) = A109613(n)-A110654(n) = A110660(n)/A110654(n). - Reinhard Zumkeller, Aug 05 2005

a(n) = 2*floor(n/2) - floor((n-1)/2). - Wesley Ivan Hurt, Oct 22 2013

a(n) = floor((n+1+(-1)^n)/2). - Wesley Ivan Hurt, Mar 15 2015

a(n) = (1 + 2n + 3(-1)^n)/4. - Wesley Ivan Hurt, Mar 18 2015

a(n) = Sum_{i=1..floor(n/2)} floor(n/(n-i)) for n > 0. - Wesley Ivan Hurt, May 21 2017

a(2n) = n+1, a(2n+1) = n, for all n >= 0. - M. F. Hasler, Oct 05 2017

EXAMPLE

1 + 2*x^2 + x^3 + 3*x^4 + 2*x^5 + 4*x^6 + 3*x^7 + 5*x^8 + 4*x^9 + 6*x^10 + 5*x^11 + ...

1 + 2*t^4 + t^6 + 3*t^8 + 2*t^10 + 4*t^12 + 3*t^14 + 5*t^16 + 4*t^18 + 6*t^20 + ...

MAPLE

series((1+x^3)/(1-x^2)^2, x, 80);

A028242:=n->floor((n+1+(-1)^n)/2): seq(A028242(n), n=0..100); # Wesley Ivan Hurt, Mar 17 2015

MATHEMATICA

lst={}; a=1; Do[a=n-a; AppendTo[lst, a], {n, 5!}]; lst (* Vladimir Joseph Stephan Orlovsky, Oct 01 2008 *)

Table[(1 + 2 n + 3 (-1)^n)/4, {n, 0, 74}] (* or *)

LinearRecurrence[{1, 1, -1}, {1, 0, 2}, 75] (* or *)

CoefficientList[Series[(1 - x + x^2)/((1 - x) (1 - x^2)), {x, 0, 74}], x] (* Michael De Vlieger, May 21 2017 *)

Table[{n, n-1}, {n, 40}]//Flatten (* Harvey P. Dale, Jun 26 2017 *)

PROG

(PARI) {a(n) = (n\2) - (n%2) + 1} \\ Michael Somos, Oct 02 1999

(PARI) A028242(n)=n\2+!bittest(n, 0) \\ M. F. Hasler, Oct 05 2017

(MAGMA) &cat[ [n+1, n]: n in [0..37] ]; // Klaus Brockhaus, Nov 23 2009

(Haskell)

import Data.List (transpose)

a028242 n = n' + 1 - m where (n', m) = divMod n 2

a028242_list = concat $ transpose [a000027_list, a001477_list]

-- Reinhard Zumkeller, Nov 27 2012

CROSSREFS

Cf. A000124 (a=1, a=n+a), A028242 (a=1, a=n-a).

Partial sums give A004652. A030451(n)=a(n+1), n>0.

Cf. A052938 (same sequence except no leading 1,0,2).

Cf. A000027, A001477.

Sequence in context: A289436 A282745 A097140 * A030451 A241825 A029162

Adjacent sequences:  A028239 A028240 A028241 * A028243 A028244 A028245

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

First part of definition adjusted to match offset by Klaus Brockhaus, Nov 23 2009

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified February 19 02:06 EST 2018. Contains 299330 sequences. (Running on oeis4.)