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%I #27 Oct 23 2019 16:26:25
%S 1,2,4,6,10,16,26,42,70,110,194,288,550,754,1586,1974,4630,5168,13634,
%T 13530,40390,35422,120146,92736,358390,242786,1071074,635622,3205030,
%U 1664080,9598706,4356618,28763350,11405774,86224514,29860704,258542470,78176338
%N Number of binary words of length n such that in every prefix and in every suffix the number of 0's and the number of 1's differ by at most two.
%C All terms with index n > 0 are even.
%H Alois P. Heinz, <a href="/A306293/b306293.txt">Table of n, a(n) for n = 0..4193</a>
%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (0,8,0,-22,0,23,0,-6).
%F G.f.: -(x+1)*(4*x^7-4*x^6-7*x^5-5*x^4+5*x^3+5*x^2-x-1) / ((3*x^2-1) *(2*x^2-1) *(x^2+x-1) *(x^2-x-1)).
%F a(n) <= A306306(n).
%e a(3) = 6: 001, 010, 011, 100, 101, 110.
%e a(4) = 10: 0010, 0011, 0100, 0101, 0110, 1001, 1010, 1011, 1100, 1101.
%e a(5) = 16: 00101, 00110, 01001, 01010, 01011, 01100, 01101, 01110, 10001, 10010, 10011, 10100, 10101, 10110, 11001, 11010.
%e a(6) = 26: 001010, 001011, 001100, 001101, 001110, 010010, 010011, 010100, 010101, 010110, 011001, 011010, 011100, 100011, 100101, 100110, 101001, 101010, 101011, 101100, 101101, 110001, 110010, 110011, 110100, 110101.
%e a(7) = 42: 0010101, 0010110, 0011001, ..., 1100110, 1101001, 1101010.
%e a(8) = 70: 00101010, ..., 00111100, ..., 11000011, ..., 11010101.
%p a:= n-> `if`(n<2, 1+n, 2*(<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>,
%p <-6|23|-22|8>>^iquo(n-2, 2, 'r').[<<2, 5, 13, 35>>,
%p <<3, 8, 21, 55>>][1+r])[1, 1]):
%p seq(a(n), n=0..50);
%Y Bisections of a(n+2)/2 give: A007689 (even part), A001906(n+2) (odd part).
%Y Cf. A068911, A128588, A303696, A306306, A306315.
%K nonn,easy
%O 0,2
%A _Alois P. Heinz_, Feb 04 2019
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