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a(n) = 2^n - n^2 - n.
1

%I #29 Aug 16 2017 17:13:47

%S 1,0,-2,-4,-4,2,22,72,184,422,914,1916,3940,8010,16174,32528,65264,

%T 130766,261802,523908,1048156,2096690,4193798,8388056,16776616,

%U 33553782,67108162,134216972,268434644,536870042,1073740894,2147482656,4294966240,8589933470,17179867994

%N a(n) = 2^n - n^2 - n.

%H Colin Barker, <a href="/A220588/b220588.txt">Table of n, a(n) for n = 0..1000</a>

%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*a(n - 1) + ((n - 3)^2 + 3(n - 3)) = 2*a(n - 1) + A028552(n - 3) for n > 4.

%F a(n) = (2*a(n-1) + 7*a(n-2))*2 = A015519/2 for n > 4.

%F From _Colin Barker_, Aug 16 2017: (Start)

%F G.f.: (1 - 5*x + 7*x^2 - x^3) / ((1 - x)^3*(1 - 2*x)).

%F a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4) for n>3.

%F (End)

%e a(3) = -4 because 2^3 - 3^2 - 3 = 8 - 9 - 3 = -4.

%e a(4) = -4 because 2^4 - 4^2 - 4 = 16 - 16 - 4 = -4.

%e a(5) = 2 because 2^5 - 5^2 - 5 = 32 - 25 - 5 = 2.

%e a(6) = 22 because 2^6 - 6^2 - 6 = 64 - 36 - 6 = 22.

%t Table[2^n - n^2 - n, {n, 0, 32}] (* _Alonso del Arte_, Dec 16 2012 *)

%o (Maxima) A220588(n):=2^n-n^2-n$ makelist(A220588(n),n,0,20); /* _Martin Ettl_, Dec 18 2012 */

%o (PARI) Vec((1 - 5*x + 7*x^2 - x^3) / ((1 - x)^3*(1 - 2*x)) + O(x^40)) \\ _Colin Barker_, Aug 16 2017

%Y Cf. A015519, A024012, A001580.

%K sign,easy

%O 0,3

%A _Dario Piazzalunga_, Dec 16 2012

%E a(3) corrected by _Charles A. Dagino_, Aug 16 2017