A008621 Expansion of 1/((1-x)*(1-x^4)).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

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

a(n-1) is the minimum independence number over all planar graphs with n vertices. The bound follows from the Four Color Theorem. It is attained by a union of 4-cliques. Other extremal graphs are examined in the Bickle link. - Allan Bickle, Feb 04 2022

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
• Allan Bickle, Independence Number of Maximal Planar Graphs, Congr. Num. 234 (2019) 61-68.
• 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).

Crossrefs

Cf. A002265 (equals this - 1).

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

Programs

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 *)

PARI

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

Python

[n//4+1 for n in range(85)] # Gennady Eremin, Mar 01 2022

Formula

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

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

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

From R. J. Mathar, Jun 04 2021: (Start)

G.f.: 1 / ( (1+x)*(1+x^2)*(x-1)^2 ).

a(n) + a(n-1) = A004524(n+3).

a(n) + a(n-2) = A008619(n). (End)

a(n) = A002265(n) + 1. - M. F. Hasler, Oct 17 2022

Extensions

More terms from Stefan Steinerberger, Apr 03 2006