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Search: id:A093709
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| A093709 |
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Characteristic function of squares or twice squares. |
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+0
5
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| 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0
(list; graph; listen)
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OFFSET
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0,1
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COMMENT
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Euler transform of period 8 sequence [1,0,-1,1,-1,0,1,-1,...].
G.f. A(x) satisfies A(x^2)=(A(x)+A(-x))/2. a(2n)=a(n).
G.f. A(x) satisfies 0=f(A(x^2)/A(x),A(x^4)/A(x^2)) where f(u,v)=u^2+v-2(u+u^2)v+2(uv)^2.
G.f. A(x) gives A(x)/A(x^2)-1=x*A092869(x^2).
Partial sums of a(n) for n >= 1 are A071860(n+1). [From Jaroslav Krizek (jaroslav.krizek(AT)atlas.cz), Oct 18 2009]
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REFERENCES
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John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156163. [From N. J. A. Sloane, (njas(AT)research.att.com), Feb 23 2009]
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LINKS
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Index entries for characteristic functions
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FORMULA
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Multiplicative with a(0)=a(2^e)=1, a(p^e)=1 if e even, 0 otherwise.
G.f.: (theta_3(q)+theta_3(q^2))/2 = 1+(Sum_{k>0} x^(k^2)+x^(2k^2)).
Dirichlet g.f.: zeta(2s)(1+2^-s).
Expansion of psi(q^4)*f(-q^3,-q^5)/f(-q,-q^7) in powers of q where psi(),f() are Ramanujan theta functions.
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PROGRAM
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(PARI) a(n)=issquare(n)|issquare(2*n)
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CROSSREFS
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A053866(n)=a(n) if n>0. Characteristic function of A028982 union 0.
Sequence in context: A116915 A076141 A011751 this_sequence A079295 A088025 A082416
Adjacent sequences: A093706 A093707 A093708 this_sequence A093710 A093711 A093712
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KEYWORD
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nonn,mult
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AUTHOR
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Michael Somos, Apr 11 2004
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