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a(n) = (n-4)*(n-3)*2^(n-2).
3

%I #41 Nov 11 2024 22:28:51

%S 3,3,2,0,0,16,96,384,1280,3840,10752,28672,73728,184320,450560,

%T 1081344,2555904,5963776,13762560,31457280,71303168,160432128,

%U 358612992,796917760,1761607680,3875536896,8489271296,18522046464,40265318400,87241523200,188441690112

%N a(n) = (n-4)*(n-3)*2^(n-2).

%C Binomial transform of (3, 0, -1, followed by A005563).

%C The sequence and its successive differences are:

%C 3, 3, 2, 0, 0, 16, 96, 384, a(n),

%C 0, -1, -2, 0, 16, 80, 288, 896, A178987,

%C -1, -1, 2, 16, 64, 208, 608, 2688, -A127276,

%C 0, 3, 14, 48, 144, 400, 1056, 2688, A176027,

%C 3, 11, 34, 96, 256, 656, 1632, 3968, A084266(n+1)

%C 8, 23, 62, 160, 400, 976, 2336, 5504,

%C 15, 39, 98, 240, 576, 1360, 3168, 7296.

%C Division of the k-th column by abs(A174882(k)) gives

%C 3, 3, 1, 0, 0, 1, 3, 3, 5, 15, 21, 14, A064038(n-3),

%C 0, -1, -1, 0, 1, 5, 9, 7, 10, 27, 35, 22, A160050(n-3),

%C -1, -1, 1, 2, 4, 13, 19, 13, 17, 43, 53, 32, A176126,

%C 0, 3, 7, 6, 9, 25, 33, 21, 26, 63, 75, 44, A178242,

%C 3, 11, 17, 12, 16, 41, 51, 31, 37, 87, 101, 58,

%C 8 23, 31, 20, 25, 61, 73, 43, 50, 115, 131, 74,

%C 15, 39, 49, 30, 36, 85, 99, 57, 65, 147, 165, 92.

%C Columns are (or from) A005563, A142463, A056220, A002378, A000290, A001844, A058331, A002061, A002522, A097080, A093328, A014206.

%H G. C. Greubel, <a href="/A181407/b181407.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (6,-12,8).

%F a(n) = 16*A001788(n-4).

%F a(n+1) - a(n) = A178987(n).

%F G.f.: (3 - 15*x + 20*x^2) / (1-2*x)^3. - _R. J. Mathar_, Jan 30 2011

%F E.g.f.: (x^2 - 3*x + 3)*exp(2*x). - _G. C. Greubel_, Feb 21 2019

%t Table[(n-4)*(n-3)*2^(n-2), {n,0,40}] (* _G. C. Greubel_, Feb 21 2019 *)

%o (Magma) [(n-4)*(n-3)*2^(n-2): n in [0..40] ]; // _Vincenzo Librandi_, Feb 01 2011

%o (PARI) vector(40, n, n--; (n-4)*(n-3)*2^(n-2)) \\ _G. C. Greubel_, Feb 21 2019

%o (Sage) [(n-4)*(n-3)*2^(n-2) for n in (0..40)] # _G. C. Greubel_, Feb 21 2019

%o (GAP) List([0..40], n-> (n-4)*(n-3)*2^(n-2)); # _G. C. Greubel_, Feb 21 2019

%Y Cf. A176027, A181318.

%K nonn,easy

%O 0,1

%A _Paul Curtz_, Jan 28 2011