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%I #24 Sep 08 2022 08:45:50
%S 1,3,11,23,51,103,211,423,851,1703,3411,6823,13651,27303,54611,109223,
%T 218451,436903,873811,1747623,3495251,6990503,13981011,27962023,
%U 55924051,111848103,223696211,447392423,894784851,1789569703,3579139411
%N a(n) = (5*2^n - 2*(-1)^n - 9)/3.
%C The sequence and higher-order differences in subsequent rows are
%C 1, 3, 11, 23, 51, 103, 211, 423, 851, 1703, 3411, 6823, 13651
%C 2, 8, 12, 28, 52, 108, 212, 428, 852, 1708, 3412, 6828, 13652
%C 6, 4, 16, 24, 56, 104, 216, 424, 856, 1704, 3416, 6824, 13656
%C -2, 12, 8, 32, 48, 112, 208, 432, 848, 1712, 3408, 6832, 13648
%C 14, -4, 24, 16, 64, 96, 224, 416, 864, 1696, 3424, 6816, 13664
%C -18, 28, -8, 48, 32, 128, 192, 448, 832, 1728, 3392, 6848, 1363
%C 46, -36, 56, -16, 96, 64, 256, 384, 896, 1664, 3456, 6784, 1369
%C The main diagonal 1,8,16,... is essentially A000079.
%C A subdiagonal is 2, 4, 8, 16, ... A155559.
%C Other diagonals are 3, 12, 24, 48, ... = 3*A151821, 6, 12, 24, ... = A082505 and -2, -4, -8, -16, ..., a negated variant of A171449.
%H Vincenzo Librandi, <a href="/A173078/b173078.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (2,1,-2).
%F a(n) = 2*a(n-1) + a(n-2) - 2*a(n-3).
%F a(n+1) - 2*a(n) = A010686(n-1).
%F a(n) = A084214(n+1) - 3.
%F G.f.: x*(1 + x + 4*x^2) / ( (1-x)*(1-2*x)*(1+x) ).
%F a(2n+3) - a(2n+1) = 10*A000302(n).
%F E.g.f.: (-2*exp(-x) + 6 - 9*exp(x) + 5*exp(2*x))/3. - _G. C. Greubel_, Dec 01 2019
%p seq( (5*2^n -2*(-1)^n -9)/3, n=1..40); # _G. C. Greubel_, Dec 01 2019
%t LinearRecurrence[{2,1,-2},{1,3,11},40] (* _Harvey P. Dale_, Oct 01 2018 *)
%o (Magma) [5*2^n/3-2*(-1)^n/3-3: n in [1..40]]; // _Vincenzo Librandi_, Aug 05 2011
%o (PARI) vector(40, n, (5*2^n - 2*(-1)^n - 9)/3) \\ _G. C. Greubel_, Dec 01 2019
%o (Sage) [(5*2^n - 2*(-1)^n - 9)/3 for n in (1..40)] # _G. C. Greubel_, Dec 01 2019
%o (GAP) List([1..40], n-> (5*2^n - 2*(-1)^n - 9)/3); # _G. C. Greubel_, Dec 01 2019
%K nonn,easy
%O 1,2
%A _Paul Curtz_, Feb 09 2010