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A255347
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a(n) = n * (1 - (-1)^(n/4) / 4) if n divisible by 4, a(n) = n otherwise.
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1
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0, 1, 2, 3, 5, 5, 6, 7, 6, 9, 10, 11, 15, 13, 14, 15, 12, 17, 18, 19, 25, 21, 22, 23, 18, 25, 26, 27, 35, 29, 30, 31, 24, 33, 34, 35, 45, 37, 38, 39, 30, 41, 42, 43, 55, 45, 46, 47, 36, 49, 50, 51, 65, 53, 54, 55, 42, 57, 58, 59, 75, 61, 62, 63, 48, 65, 66
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OFFSET
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0,3
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LINKS
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Index entries for linear recurrences with constant coefficients, signature (2,-1,0,-2,4,-2,0,-1,2,-1).
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FORMULA
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Euler transform of length 10 sequence [2, 0, 1, -2, 1, -1, 0, 2, 0, -1].
a(n) is multiplicative with a(2) = 2, a(4) = 5, a(2^e) = 3*2^(e-2) if e>2, a(p^e) = p^e otherwise. [corrected by Amiram Eldar, Dec 29 2022]
G.f.: f(x) - f(-x^4) where f(x) := x / (1 - x)^2.
G.f.: x * (1 + x^3) * (1 + x^5) / ((1 - x)^2 * (1 + x^4)^2).
a(n) = -a(-n) for all n in Z.
Dirichlet g.f.: zeta(s-1)*(1+1/4^s-4/8^s). - Amiram Eldar, Dec 29 2022
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EXAMPLE
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G.f. = x + 2*x^2 + 3*x^3 + 5*x^4 + 5*x^5 + 6*x^6 + 7*x^7 + 6*x^8 + 9*x^9 + ...
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MATHEMATICA
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a[ n_] := n {1, 1, 1, 5/4, 1, 1, 1, 3/4}[[Mod[ n, 8, 1]]];
a[ n_] := n If[ Divisible[ n, 4], 1 - (-1)^(n/4) / 4, 1];
LinearRecurrence[{2, -1, 0, -2, 4, -2, 0, -1, 2, -1}, {0, 1, 2, 3, 5, 5, 6, 7, 6, 9}, 70] (* Harvey P. Dale, Jul 28 2018 *)
CoefficientList[Series[x*(1+x^3)*(1+x^5)/((1-x)^2*(1+x^4)^2), {x, 0, 60}], x] (* G. C. Greubel, Aug 02 2018 *)
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PROG
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(PARI) {a(n) = n * if( n%4, 1, 1 - (-1)^(n/4) / 4)};
(PARI) {a(n) = n * [3/4, 1, 1, 1, 5/4, 1, 1, 1][n%8 + 1]};
(PARI) my(x='x+O('x^60)); concat([0], Vec(x*(1+x^3)*(1+x^5)/((1-x)^2*(1 + x^4)^2))) \\ G. C. Greubel, Aug 02 2018
(Magma) m:=60; R<x>:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x^3)*(1+x^5)/((1-x)^2*(1+x^4)^2))); // G. C. Greubel, Aug 02 2018
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CROSSREFS
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KEYWORD
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nonn,mult,easy
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AUTHOR
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STATUS
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approved
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