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A176447
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a(2n) = -n, a(2n+1) = 2n+1.
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2
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0, 1, -1, 3, -2, 5, -3, 7, -4, 9, -5, 11, -6, 13, -7, 15, -8, 17, -9, 19, -10, 21, -11, 23, -12, 25, -13, 27, -14, 29, -15, 31, -16, 33, -17, 35, -18, 37, -19, 39, -20, 41, -21, 43, -22, 45, -23, 47, -24, 49, -25, 51, -26, 53, -27, 55, -28, 57, -29, 59, -30, 61, -31, 63, -32, 65, -33, 67, -34, 69, -35
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OFFSET
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0,4
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COMMENTS
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There is more complicated way of defining the sequence: consider the sequence of modified Bernoulli numbers EVB(n) = A176327(n)/A176289(n) and its inverse binomial transform IEVB(n) = A176328(n)/A176591(n). Then a(n) is the numerator of the difference EVB(n)-IEVB(n). The denominator of the difference is 1 if n=0, else A040001(n-1).
A particularity of EVB(n) is: its (forward) binomial transform is 1, 1, 7/6, 3/2, 59/30,.. = (-1)^n*IEVB(n).
Note that A026741 is related to the Rydberg-Ritz spectrum of the hydrogen atom.
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LINKS
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FORMULA
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a(n) = (-1)^n*A026741(n) = n*(1-3*(-1)^n)/4.
G.f.: x*(1-x+x^2) / ( (x-1)^2*(1+x)^2 ).
a(n) = +2*a(n-2) -a(n-4). (End)
Euler transform of length 6 sequence [ -1, 3, 1, 0, 0, -1].
0 = - 1 - a(n) - a(n+1) + a(n+2) + a(n+3) for all n in Z.
0 = 1 + a(n)*(-2 -a(n) + a(n+2)) - 2*a(n+1) - a(n+2) for all n in Z. (End)
a(n) is multiplicative with a(2^e) = -(2^(e-1)) if e>0, a(p^e) = p^e otherwise.
G.f.: (f(x) - 3 * f(-x)) / 4 where f(x) := x / (1 - x)^2.
G.f.: x * (1 - x) * (1 - x^6) / ((1 - x^2)^3 * (1 - x^3)). (End)
Dirichlet g.f.: zeta(s-1) * (1 - 3/2^s).
Sum_{k=0..n} a(k) = A008795(n-1), for n > 0.
Sum_{k=0..n} a(k) ~ n^2/8. (End)
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EXAMPLE
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G.f. = x - x^2 + 3*x^3 - 2*x^4 + 5*x^5 - 3*x^6 + 7*x^7 - 4*x^8 + 9*x^9 - 5*x^10 + ...
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MATHEMATICA
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a[n_?EvenQ]:=-(n/2); a[n_?OddQ]:=n; Table[a[n], {n, 100}] (* Alonso del Arte, Dec 01 2010 *)
a[ n_] := n / If[ Mod[ n, 2] == 1, 1, -2]; (* Michael Somos, Jun 11 2013 *)
CoefficientList[Series[x (1 - x + x^2)/((x - 1)^2*(1 + x)^2), {x, 0, 70}], x] (* Michael De Vlieger, Dec 10 2016 *)
LinearRecurrence[{0, 2, 0, -1}, {0, 1, -1, 3}, 80] (* Harvey P. Dale, Nov 01 2017 *)
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PROG
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(PARI) {a(n) = n / if( n%2, 1, -2)}; /* Michael Somos, Jun 11 2013 */
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CROSSREFS
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KEYWORD
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sign,easy,mult
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AUTHOR
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STATUS
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approved
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