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A259966 Total binary weight (cf. A000120) of all A005251(n) binary sequences of length n not containing any isolated 1's.

%I A259966 #35 Mar 24 2020 17:39:27
%S A259966 0,0,2,7,16,34,72,149,300,593,1158,2239,4292,8168,15450,29072,54456,
%T A259966 101597,188878,350038,646880,1192415,2192956,4024583,7371884,13479421,
%U A259966 24607048,44853552,81645236,148424000,269497614,488784787,885571340,1602879242,2898512344
%N A259966 Total binary weight (cf. A000120) of all A005251(n) binary sequences of length n not containing any isolated 1's.
%D A259966 R. K. Guy, Letter to N. J. A. Sloane, Feb 05 1986.
%H A259966 Reinhard Zumkeller, <a href="/A259966/b259966.txt">Table of n, a(n) for n = 0..1000</a>
%H A259966 Steven Finch, <a href="https://arxiv.org/abs/2003.09458">Cantor-solus and Cantor-multus distributions</a>, arXiv:2003.09458 [math.CO], 2020.
%H A259966 R. K. Guy, <a href="/A005251/a005251.pdf">Letter to N. J. A. Sloane, Feb 1986</a>
%H A259966 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-6,6,-5,2,-1).
%F A259966 a(n) = a(n-1)+a(n-2)+2*b(n)+a(n-4)+3*b(n-2), where b() is A005251().
%F A259966 G.f.: -x^2*(x-2) / (x^3-x^2+2*x-1)^2. - _Colin Barker_, Jul 21 2015
%F A259966 a(n) = Sum_{k=1..n} k * A097230(n,k). - _Alois P. Heinz_, Mar 03 2020
%e A259966 The only two 2-bitstrings without isolated 1's are 00 and 11.  The bitsums of these are 0 and 2.  Adding these give a(2)=2.
%e A259966 The only four 3-bitstrings without isolated 1's are 000, 011, 110 and 111.  The bitsums of these are 0, 2, 2 and 3.  Adding these give a(3)=7.
%o A259966 (Haskell)
%o A259966 a259966 n = a259966_list !! n
%o A259966 a259966_list = 0 : 0 : 2 : 7 : zipWith (+)
%o A259966    (zipWith3 (((+) .) . (+))
%o A259966              a259966_list (drop 2 a259966_list) (drop 3 a259966_list))
%o A259966    (drop 2 $ zipWith (+)
%o A259966              (map (* 2) $ drop 2 a005251_list) (map (* 3) a005251_list))
%o A259966 -- _Reinhard Zumkeller_, Jul 13 2015
%o A259966 (PARI) concat([0,0], Vec(-x^2*(x-2)/(x^3-x^2+2*x-1)^2 + O(x^50))) \\ _Colin Barker_, Jul 21 2015
%Y A259966 Cf. A005251, A097230.
%K A259966 nonn,easy
%O A259966 0,3
%A A259966 _N. J. A. Sloane_, Jul 11 2015
%E A259966 Edited by _Reinhard Zumkeller_, Jul 13 2015