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%I #18 Mar 23 2024 07:23:03
%S 1,2,3,6,10,18,29,49,78,128,203,329,523,844,1347,2172,3480,5614,9023,
%T 14567,23466,37910,61165,98865,159677,258190,417283,674890,1091214,
%U 1765146,2854793,4618373,7470614,12086436,19552903,31635193,51181367,82809832
%N Number of binary strings u of any length with property that length(u) + number of 0's in u <= n (only one of a string and its reversal are counted).
%C If 0 is replaced by 2 (as in A007931) "length + 0-bits" is simply the total of ternary digits (e.g., 3 for 21 instead of 01).
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-4,4,-2,1,1,-1)
%F G.f.: x*(-x^7-x^4+3x^3-2x^2-x+1)/((1-x-x^2)*(1-x^2-x^4)*(1-x)^2).
%e a(3) = 3: 0 01 111 (e.g. 01: length 2 + 1 zero = 3).
%e a(4) = 6: 0 01 00 011 101 1111.
%e a(5) =10: 0 01 00 011 101 001 010 0111 1011 11111.
%t CoefficientList[Series[x (-x^7-x^4+3x^3-2x^2-x+1)/((1-x-x^2) (1-x^2-x^4) (1-x)^2),{x,0,50}],x] (* _Harvey P. Dale_, Jun 15 2011 *)
%Y If reversals are counted as distinct then we obtain A000126.
%Y A007931 (binary strings represented by ternary numbers),
%Y Cf. A035615 (binary "same game").
%K nonn
%O 1,2
%A _Frank Ellermann_, Dec 02 2001
%E More terms from _Harvey P. Dale_, Jun 15 2011
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