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A008621 Expansion of 1/((1-x)*(1-x^4)). 22
1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 17, 17, 17, 17, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Arises from Gleason's theorem on self-dual codes: 1/((1-x^2)*(1-x^8)) is the Molien series for the real 2-dimensional Clifford group (a dihedral group of order 16) of genus 1.

Thickness of the hypercube graph Q_n. - Eric W. Weisstein, Sep 09 2008

Count of odd numbers between consecutive quarter-squares, A002620. Oppermann's conjecture states that for each count there will be at least one prime. - Fred Daniel Kline, Sep 10 2011

Number of partitions into parts 1 and 4. - Joerg Arndt, Jun 01 2013

REFERENCES

D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.

F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes, 1977, Chapter 19, Problem 3, p. 602.

LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 211

G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.

Eric Weisstein's World of Mathematics, Graph Thickness

Wikipedia, Oppermann's Conjecture

Index entries for Molien series

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

FORMULA

a(n) = floor(n/4) + 1.

a(n) = A010766(n+4, 4).

a(n) = {Sum{k=0..n, (k+1)cos(pi*(n-k)/2}+1/4[cos(n*Pi/2)+1+(-1)^n] }/2 - Paolo P. Lava, Oct 09 2006

Also, a(n) = ceiling((n+1)/4), n >= 0. - Mohammad K. Azarian, May 22 2007

a(n) = Sum_{i=0..n} A121262(i) = n/4 +5/8 +(-1)^n/8 + A057077(n)/4. - R. J. Mathar, Mar 14 2011

a(x,y):= floor(x/2) + floor(y/2) - x where x=A002620(n) and y=A002620(n+1), n>2. - Fred Daniel Kline, Sep 10 2011

a(0)=1, a(1)=1, a(2)=1, a(3)=1, a(4)=2, a(n) = a(n-1) + a(n-4) - a(n-5). - Harvey P. Dale, Feb 19 2012

G.f.: 1 / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Jun 04 2021

a(n)+a(n-1) = A004524(n+3). a(n)+a(n-2)=A008619(n). - R. J. Mathar, Jun 04 2021

MATHEMATICA

Table[Floor[n/4]+1, {n, 0, 80}] (* Stefan Steinerberger, Apr 03 2006 *)

CoefficientList[Series[1/((1-x)(1-x^4)), {x, 0, 80}], x] (* Harvey P. Dale, Feb 19 2012 *)

Flatten[ Table[ PadRight[{}, 4, n], {n, 19}]] (* Harvey P. Dale, Feb 19 2012 *)

PROG

(PARI) a(n)=n\4+1 \\ Charles R Greathouse IV, Feb 06 2017

CROSSREFS

Cf. A008718, A024186, A110160, A110868, A110869, A110876, A110880, A002265, A008620.

Sequence in context: A002265 A242601 A110655 * A144075 A128929 A257839

Adjacent sequences:  A008618 A008619 A008620 * A008622 A008623 A008624

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Stefan Steinerberger, Apr 03 2006

STATUS

approved

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Last modified September 18 11:01 EDT 2021. Contains 347518 sequences. (Running on oeis4.)