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A306293 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. 4
1, 2, 4, 6, 10, 16, 26, 42, 70, 110, 194, 288, 550, 754, 1586, 1974, 4630, 5168, 13634, 13530, 40390, 35422, 120146, 92736, 358390, 242786, 1071074, 635622, 3205030, 1664080, 9598706, 4356618, 28763350, 11405774, 86224514, 29860704, 258542470, 78176338 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

All terms with index n > 0 are even.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..4193

Index entries for linear recurrences with constant coefficients, signature (0,8,0,-22,0,23,0,-6).

FORMULA

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)).

a(n) <= A306306(n).

EXAMPLE

a(3) = 6: 001, 010, 011, 100, 101, 110.

a(4) = 10: 0010, 0011, 0100, 0101, 0110, 1001, 1010, 1011, 1100, 1101.

a(5) = 16: 00101, 00110, 01001, 01010, 01011, 01100, 01101, 01110, 10001, 10010, 10011, 10100, 10101, 10110, 11001, 11010.

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.

a(7) = 42: 0010101, 0010110, 0011001, ..., 1100110, 1101001, 1101010.

a(8) = 70: 00101010, ..., 00111100, ..., 11000011, ..., 11010101.

MAPLE

a:= n-> `if`(n<2, 1+n, 2*(<<0|1|0|0>, <0|0|1|0>, <0|0|0|1>,

        <-6|23|-22|8>>^iquo(n-2, 2, 'r').[<<2, 5, 13, 35>>,

        <<3, 8, 21, 55>>][1+r])[1, 1]):

seq(a(n), n=0..50);

CROSSREFS

Bisections of a(n+2)/2 give: A007689 (even part), A001906(n+2) (odd part).

Cf. A068911, A128588, A303696, A306306, A306315.

Sequence in context: A094985 A128588 A023613 * A307795 A065795 A293077

Adjacent sequences:  A306290 A306291 A306292 * A306294 A306295 A306296

KEYWORD

nonn,easy

AUTHOR

Alois P. Heinz, Feb 04 2019

STATUS

approved

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Last modified March 29 17:23 EDT 2020. Contains 333116 sequences. (Running on oeis4.)