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

0,5

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

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).

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

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

Crossrefs

Cf. A002265 (equals this - 1).

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