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A008621
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Expansion of 1/((1-x)*(1-x^4)).
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19
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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
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OFFSET
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0,5
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COMMENTS
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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.
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]
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REFERENCES
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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.
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LINKS
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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.
Index entries for Molien series
Index to sequences with linear recurrences with constant coefficients, signature (1,0,0,1,-1).
Wikipedia,Oppermann's Conjecture
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FORMULA
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a(n)= floor((n+3)/4), n>0;
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/4), n>=1. - 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 [From 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) [From Harvey P. Dale, Feb 19 2012]
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MATHEMATICA
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Table[Floor[(n + 3)/4], {n, 1, 80}] - Stefan Steinerberger, Apr 03 2006
CoefficientList[Series[1/((1-x)(1-x^4)), {x, 0, 80}], x] (* or *) Flatten[ Table[ PadRight[{}, 4, n], {n, 19}]] (* From Harvey P. Dale, Feb 19 2012 *)
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CROSSREFS
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Cf. A008718, A024186, A110160, A110868, A110869, A110876, A110880.
Cf. A008620, A002265.
a(n)=A010766(n+4, 4).
Sequence in context: A056172 A091373 A197637 * A002265 A110655 A144075
Adjacent sequences: A008618 A008619 A008620 * A008622 A008623 A008624
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Stefan Steinerberger, Apr 03 2006
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STATUS
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approved
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