A236291 Number of length n binary words that contain an even number of 0's or exactly two 1's.

1, 1, 2, 7, 8, 26, 32, 85, 128, 292, 512, 1079, 2048, 4174, 8192, 16489, 32768, 65672, 131072, 262315, 524288, 1048786, 2097152, 4194557, 8388608, 16777516, 33554432, 67109215, 134217728, 268435862, 536870912, 1073742289, 2147483648, 4294967824, 8589934592

Offset

0,3

Links

Table of n, a(n) for n=0..34.

Index entries for linear recurrences with constant coefficients, signature (2,3,-6,-3,6,1,-2).

Formula

G.f.: (1 - x - 3*x^2 + 6*x^3 - 3*x^4 - 2*x^5 - 3*x^6 + x^7)/( (1 - 2*x)*(1 - x^2)^3 ).

a(n) = (2^(1+n))/4 for n even; a(n) = (2^(1+n)-2*n+2*n^2)/4 for n odd. - Colin Barker, Jan 23 2014

E.g.f.: (1 + cosh(2*x) + x^2*sinh(x) + sinh(2*x))/2. - Stefano Spezia, Mar 20 2022

Example

a(3)=7 because we have: 001, 010, 011, 100, 101, 110, 111.

Mathematica

nn=30; CoefficientList[Series[(1-x-3*x^2+6*x^3-3*x^4-2*x^5-3*x^6+x^7)/ ((1-2*x)*(1-x^2)^3), {x, 0, nn}], x]
LinearRecurrence[{2, 3, -6, -3, 6, 1, -2}, {1, 1, 2, 7, 8, 26, 32, 85}, 40] (* Harvey P. Dale, Dec 18 2022 *)

Crossrefs

Cf. A161680 (words containing exactly two 1's), A011782 (words containing an even number of 0's), A000384 (words containing an even number of 0's and exactly 2 1's).

Sequence in context: A291629 A117558 A117559 * A190175 A081700 A093795

Adjacent sequences: A236288 A236289 A236290 * A236292 A236293 A236294

Keyword

nonn,easy

Author

Geoffrey Critzer, Jan 21 2014

Extensions

More terms from Colin Barker, Jan 23 2014

Status

approved