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%I #29 Feb 06 2023 13:33:11
%S 0,0,1,3,7,14,30,62,126,253,509,1021,2045,4092,8188,16380,32764,65531,
%T 131067,262139,524283,1048570,2097146,4194298,8388602,16777209,
%U 33554425,67108857,134217721,268435448,536870904,1073741816,2147483640
%N a(n) = round((2^n - n - 1)/4).
%C The variance v(n) = Sum_{k=0..2^n-n-1} (k - m(n))^2*p(n,k) of the distribution function p(n,k) = binomial(2^n -n-1, k)/2^(2^n -n-1) with m(n) its mean value is 0., 0.25, 1., 2.75, 6.5, 14.25, 30., 61.75, 125.5, 253.25, 509., 1020.75, 2044.5, 4092.25, 8188... We set a(n) = round(v(n)).
%H G. C. Greubel, <a href="/A173010/b173010.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,0,1,-3,2).
%F a(n) = round((2^n -n -1)/4).
%F G.f.: x^3*(1 -x^3 +x^4)/(1 -3*x +2*x^2 -x^4 +3*x^5 -2*x^6). [sign corrected by _Georg Fischer_, Apr 17 2020]
%F v(n) = (1/8)*2^n -1/4 + v(-1+n) with v(1) = 0 and a(n) = round(v(n)).
%F a(n) = round(A000295(n)/4). - _G. C. Greubel_, Feb 20 2021
%p A173010:= round((2^n -n-1)/4); seq(A173010(n), n=1..40); # _G. C. Greubel_, Feb 20 2021
%t nn:=33; Rest[CoefficientList[Series[x^3*(1-x^3+x^4)/(1-3*x+2*x^2-2*x^6-x^4+3*x^5),{x,0,nn}],x]] (* _Georg Fischer_, Apr 17 2020 *)
%t LinearRecurrence[{3,-2,0,1,-3,2},{0,0,1,3,7,14,30},40] (* _Harvey P. Dale_, Feb 06 2023 *)
%o (Sage) [round((2^n -n -1)/4) for n in (1..40)] # _G. C. Greubel_, Feb 20 2021
%o (Magma) [Round((2^n -n-1)/4): n in [1..40]]; // _G. C. Greubel_, Feb 20 2021
%Y Cf. A000295, A173009.
%K nonn
%O 1,4
%A _Thomas Wieder_, Feb 07 2010
%E Edited by _Georg Fischer_ and _Joerg Arndt_, Apr 17 2020
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