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A008621
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Expansion of 1/((1-x)*(1-x^4)).
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25
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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, 19, 19, 20, 20, 20, 20, 21, 21
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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
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
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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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FORMULA
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a(n) = floor(n/4) + 1.
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
G.f.: 1 / ( (1+x)*(1+x^2)*(x-1)^2 ).
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MATHEMATICA
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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 *)
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PROG
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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