%I #23 Aug 10 2017 06:55:07
%S 1,3,7,14,26,47,85,156,292,557,1079,2114,4174,8283,16489,32888,65672,
%T 131225,262315,524478,1048786,2097383,4194557,8388884,16777516,
%U 33554757,67109215,134218106,268435862,536871347,1073742289,2147484144
%N a(n) = (2^(n+1) + n^2 + n)/2.
%C Row sums of triangle A132108.
%C Binomial transform of [1, 2, 2, 1, 1, 1, ...].
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (5,-9,7,-2).
%F a(n) = (2^(n+1) + n^2 + n)/2. - _Eric W. Weisstein_, Aug 09 2017
%F a(n) = A290699(n+1)/2. - _Eric W. Weisstein_, Aug 09 2017
%F G.f.: ( 1-2*x+x^2-x^3 ) / ( (2*x-1)*(x-1)^3 ). - _R. J. Mathar_, May 23 2016
%F a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4). - _Eric W. Weisstein_, Aug 09 2017
%e a(3) = 14 = sum of row 3 terms of triangle A132108: (4 + 5 + 4 + 1).
%e a(3) = 14 = (1, 3, 3, 1) dot (1, 2, 2, 1) = (1 + 6 + 6 + 1).
%t Table[2^n + n^2 - n, {n, 20}]/2 (* _Vladimir Joseph Stephan Orlovsky_, May 07 2010 *)
%t LinearRecurrence[{5, -9, 7, -2}, {1, 3, 7, 14}, 20] (* _Eric W. Weisstein_, Aug 09 2017 *)
%t CoefficientList[Series[(1 - 2 x + x^2 - x^3)/((-1 + x)^3 (-1 + 2 x)), {x, 0, 20}], x] (* _Eric W. Weisstein_, Aug 09 2017 *)
%o (PARI) a(n) = n*(n+1)/2 + 2^n; \\ _Altug Alkan_, Aug 10 2017
%Y Cf. A132108.
%Y Cf. A290699.
%K nonn,easy
%O 0,2
%A _Gary W. Adamson_, Aug 09 2007
%E More terms from _Vladimir Joseph Stephan Orlovsky_, May 07 2010
%E Name changed to the simple formula by _Eric W. Weisstein_, Aug 09 2017