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%I #97 Sep 23 2023 03:39:56
%S 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,
%T 9,9,10,10,10,10,11,11,11,11,12,12,12,12,13,13,13,13,14,14,14,14,15,
%U 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
%N Expansion of 1/((1-x)*(1-x^4)).
%C 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.
%C Thickness of the hypercube graph Q_n. - _Eric W. Weisstein_, Sep 09 2008
%C 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
%C Number of partitions into parts 1 and 4. - _Joerg Arndt_, Jun 01 2013
%C 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
%D D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge, 1993, p. 100.
%D F. J. MacWilliams and N. J. A. Sloane, Theory of Error-Correcting Codes, 1977, Chapter 19, Problem 3, p. 602.
%H T. D. Noe, <a href="/A008621/b008621.txt">Table of n, a(n) for n = 0..1000</a>
%H Allan Bickle, <a href="https://allanbickle.files.wordpress.com/2016/05/planarindependence2.pdf">Independence Number of Maximal Planar Graphs</a>, Congr. Num. 234 (2019) 61-68.
%H INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=211">Encyclopedia of Combinatorial Structures 211</a>
%H G. Nebe, E. M. Rains and N. J. A. Sloane, <a href="http://neilsloane.com/doc/cliff2.html">Self-Dual Codes and Invariant Theory</a>, Springer, Berlin, 2006.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphThickness.html">Graph Thickness</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Oppermann%27s_conjecture">Oppermann's conjecture</a>
%H <a href="/index/Mo#Molien">Index entries for Molien series</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F a(n) = floor(n/4) + 1.
%F a(n) = A010766(n+4, 4).
%F Also, a(n) = ceiling((n+1)/4), n >= 0. - _Mohammad K. Azarian_, May 22 2007
%F a(n) = Sum_{i=0..n} A121262(i) = n/4 + 5/8 + (-1)^n/8 + A057077(n)/4. - _R. J. Mathar_, Mar 14 2011
%F 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
%F 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
%F From _R. J. Mathar_, Jun 04 2021: (Start)
%F G.f.: 1 / ( (1+x)*(1+x^2)*(x-1)^2 ).
%F a(n) + a(n-1) = A004524(n+3).
%F a(n) + a(n-2) = A008619(n). (End)
%F a(n) = A002265(n) + 1. - _M. F. Hasler_, Oct 17 2022
%t Table[Floor[n/4]+1, {n, 0, 80}] (* _Stefan Steinerberger_, Apr 03 2006 *)
%t CoefficientList[Series[1/((1-x)(1-x^4)),{x,0,80}],x] (* _Harvey P. Dale_, Feb 19 2012 *)
%t Flatten[ Table[ PadRight[{},4,n],{n,19}]] (* _Harvey P. Dale_, Feb 19 2012 *)
%o (PARI) a(n)=n\4+1 \\ _Charles R Greathouse IV_, Feb 06 2017
%o (Python) [n//4+1 for n in range(85)] # _Gennady Eremin_, Mar 01 2022
%Y Cf. A002265 (equals this - 1).
%Y Cf. A008718, A024186, A110160, A110868, A110869, A110876, A110880, A008620.
%K nonn,easy,nice
%O 0,5
%A _N. J. A. Sloane_
%E More terms from _Stefan Steinerberger_, Apr 03 2006
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