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%I #24 Apr 14 2022 01:20:57
%S 0,0,4,17,46,116,288,683,1548,3403,7320,15461,32146,65954,133800,
%T 268804,535434,1058533,2078732,4057858,7878814,15223495,29285368,
%U 56109673,107108104,203766859,386443052,730768044,1378180568,2592664120,4866008208,9112796113
%N Total binary weight squared of all A005251(n) binary sequences of length n not containing any isolated 1's.
%H G. C. Greubel, <a href="/A332863/b332863.txt">Table of n, a(n) for n = 0..1000</a>
%H Steven Finch, <a href="https://arxiv.org/abs/2003.09458">Cantor-solus and Cantor-multus distributions</a>, arXiv:2003.09458 [math.CO], 2020.
%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (6,-15,23,-27,24,-16,9,-3,1).
%F G.f.: x^2*(4-7*x+4*x^2+3*x^3-x^4)/(1-2*x+x^2-x^3)^3.
%F a(n) = Sum_{k=1..n} k^2 * A097230(n,k). - _Alois P. Heinz_, Mar 03 2020
%e The only two 2-bitstrings without isolated 1's are 00 and 11. The bitsums squared of these are 0 and 4. Adding these give a(2)=4.
%e The only four 3-bitstrings without isolated 1's are 000, 011, 110 and 111. The bitsums squared of these are 0, 4, 4 and 9. Adding these give a(3)=17.
%t LinearRecurrence[{6,-15,23,-27,24,-16,9,-3,1}, {0,0,4,17,46,116,288,683,1548}, 40] (* _G. C. Greubel_, Apr 13 2022 *)
%o (Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0,0] cat Coefficients(R!( x^2*(4-7*x+4*x^2+3*x^3-x^4)/(1-2*x+x^2-x^3)^3 )); // _G. C. Greubel_, Apr 13 2022
%o (SageMath)
%o def A332863_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( x^2*(4-7*x+4*x^2+3*x^3-x^4)/(1-2*x+x^2-x^3)^3 ).list()
%o A332863_list(40) # _G. C. Greubel_, Apr 13 2022
%Y Cf. A005251, A097230, A259966.
%K nonn,easy
%O 0,3
%A _Steven Finch_, Feb 27 2020
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