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A145390
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Number of sublattices of index n of a centered rectangular lattice fixed by a reflection.
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6
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1, 1, 2, 3, 2, 2, 2, 5, 3, 2, 2, 6, 2, 2, 4, 7, 2, 3, 2, 6, 4, 2, 2, 10, 3, 2, 4, 6, 2, 4, 2, 9, 4, 2, 4, 9, 2, 2, 4, 10, 2, 4, 2, 6, 6, 2, 2, 14, 3, 3, 4, 6, 2, 4, 4, 10, 4, 2, 2, 12, 2, 2, 6, 11, 4, 4, 2, 6, 4, 4, 2, 15, 2, 2, 6, 6, 4, 4, 2, 14, 5, 2, 2, 12, 4, 2, 4, 10, 2, 6, 4, 6, 4, 2, 4, 18, 2, 3, 6, 9, 2
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OFFSET
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1,3
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COMMENTS
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a(n) is the Dirichlet convolution of A000012 and A098178. - Domenico (domenicoo(AT)gmail.com), Oct 21 2009
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LINKS
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FORMULA
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Dirichlet g.f.: (1-2^(-s) + 2*4^(-s))*zeta^2(s).
G.f.: Sum_n (1 + cos(n*Pi/2)) x^n / (1 - x^n). - Domenico (domenicoo(AT)gmail.com), Oct 21 2009
If 4|n then a(n) = d(n) - d(n/2) + 2*d(n/4); else if 2|n then a(n) = d(n) - d(n/2); else a(n) = d(n); where d(n) is the number of divisors of n. [Rutherford] - Andrey Zabolotskiy, Mar 10 2018
Sum_{k=1..n} a(k) ~ n*(log(n) - 1 + 2*gamma - log(2)/2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 02 2019
Multiplicative with a(2^e) = 2*e-1 and a(p^e) = e+1 for an odd prime p. - Amiram Eldar, Aug 27 2023
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MAPLE
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nmax := 100 :
L := [1, -1, 0, 2, seq(0, i=1..nmax)] :
MOBIUSi(%) :
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MATHEMATICA
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m = 101; Drop[ CoefficientList[ Series[ Sum[(1 + Cos[n*Pi/2])*x^n/(1 - x^n), {n, 1, m}], {x, 0, m}], x], 1] (* Jean-François Alcover, Sep 20 2011, after formula *)
f[p_, e_] := e+1; f[2, e_] := 2*e-1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 27 2023 *)
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PROG
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(PARI) t1=direuler(p=2, 200, 1/(1-X)^2)
t2=direuler(p=2, 2, 1-X+2*X^2, 200)
t3=dirmul(t1, t2)
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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