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%I #46 Jul 15 2024 21:13:51
%S 0,1,1,-3,-3,9,9,-27,-27,81,81,-243,-243,729,729,-2187,-2187,6561,
%T 6561,-19683,-19683,59049,59049,-177147,-177147,531441,531441,
%U -1594323,-1594323,4782969,4782969,-14348907,-14348907,43046721,43046721,-129140163,-129140163,387420489
%N Expansion of g.f. (x + x^2)/(1 + 3*x^2).
%C This is the inverse binomial transform of A157241.
%C Successive differences of A157241 begin:
%C 0, 1, 3, 3, -5, -21, -21, 43, 171, 171, ... = A157241
%C 1, 2, 0, -8, -16, 0, 64, 128, 0, -512, ... = A088138
%C 1, -2, -8, -8, 16, 64, 64, -128, -512, -512, ... = A138230
%C -3, -6, 0, 24, 48, 0, -192, -384, 0, 1536, ...
%C -3, 6, 24, 24, -48, -192, -192, 384, 1536, 1536, ...
%C 9, 18, 0, -72, -144, 0, 576, 1152, 0, -4608, ...
%C 9, -18, -72 -72, 144, 576, 576, -1152, -4608, -4608, ...
%C ...
%C a(n) is the n-th term of the first column.
%C Successive differences of a(n) begin:
%C 0, 1, 1, -3, -3, 9, 9, -27, -27, 81, ...
%C 1, 0, -4, 0, 12, 0, -36, 0, 108, 0, ...
%C -1, -4, 4, 12, -12, -36, 36, 108, -108, -324, ...
%C -3, 8, 8, -24, -24, 72, 72, -216, -216, 648, ...
%C 11, 0, -32, 0, 96, 0, -288, 0, 864, 0, ...
%C -11, -32, 32, 96, -96, -288, 288, 864, -864, -2592, ...
%C -21, 64, 64, -192, -192, 576, 576, -1728, -1728, 5184, ...
%C 85, 0, -256, 0, 768, 0, -2304, 0, 6912, 0, ...
%C ...
%C First column appears to be a subsequence of Jacobsthal numbers A001045 (the trisection A082311 is missing), second column is A104538, and third column is A137717.
%C a(n) = A128019(n-2) for n > 2. - _Georg Fischer_, Oct 23 2018
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0, -3).
%F a(n) = -3*a(n-2) for n > 2.
%F E.g.f.: (1 - cos(sqrt(3)*x) + sqrt(3)*sin(sqrt(3)*x))/3. - _Stefano Spezia_, Jul 15 2024
%t Join[{0}, LinearRecurrence[{0, -3}, {1, 1}, 40]]
%t (* or, computation from b = A157241 : *)
%t b[n_] := (Switch[Mod[n, 3], 0, (-1)^((n + 3)/3), 1, (-1)^((n + 5)/3), 2, (-1)^((n + 4)/3)*2]*2^n + 1)/3; tb = Table[b[n], {n, 0, 40}]; Table[ Differences[tb, n], {n, 0, 40}][[All, 1]]
%o (PARI) concat([0], Vec((x + x^2)/(1 + 3*x^2) + O(x^40))) \\ _Felix Fröhlich_, Oct 23 2018
%Y Cf. A001045, A082311, A088138, A104538, A108411, A128019, A137717, A138230, A140429, A157241.
%K sign,easy
%O 0,4
%A _Jean-François Alcover_ and _Paul Curtz_, Jul 19 2017