A269416 Expansion of 3*(2 - x)/((1 - x)*(1 + x)^2).
6, -9, 15, -18, 24, -27, 33, -36, 42, -45, 51, -54, 60, -63, 69, -72, 78, -81, 87, -90, 96, -99, 105, -108, 114, -117, 123, -126, 132, -135, 141, -144, 150, -153, 159, -162, 168, -171, 177, -180, 186, -189, 195, -198, 204, -207, 213, -216, 222, -225, 231, -234, 240
Offset: 0
Examples
a(0) = 1 + 2 + 3 = 6; a(1) = 1 + 2 + 3 - 4 - 5 - 6 = -9; a(2) = 1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 = 15; a(3) = 1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 - 10 - 11 - 12 = -18; a(4) = 1 + 2 + 3 - 4 - 5 - 6 + 7 + 8 + 9 - 10 - 11 - 12 + 13 + 14 + 15 = 24, etc.
Links
- Index entries for linear recurrences with constant coefficients, signature (-1,1,1).
Programs
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Mathematica
LinearRecurrence[{-1, 1, 1}, {6, -9, 15}, 53] Table[3 ((6 (-1)^n n + 7 (-1)^n + 1)/4), {n, 0, 52}]
Formula
G.f.: 3*(2 - x)/((1 - x)*(1 + x)^2).
a(n) = -a(n-1) + a(n-2) + a(n-3).
a(n) = Sum_{k=0..n} (-1)^k*3*(3*k + 2).
a(n) = 3*((-1)^n*6*n + (-1)^n*7 + 1)/4.
Sum_{n>=0} 1/a(n) = log(3)/6 - Pi/(18*sqrt(3)) = 0.082335416765006179088425414... . - Vaclav Kotesovec, Feb 25 2016
a(n) = 3*(-1)^n*A007494(n+1). - R. J. Mathar, Jun 07 2016
Comments