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A081374 Size of "uniform" Hamming covers of distance 1, that is, Hamming covers in which all vectors of equal weight are treated the same, included or excluded from the cover together. 5
1, 2, 2, 5, 10, 22, 43, 86, 170, 341, 682, 1366, 2731, 5462, 10922, 21845, 43690, 87382, 174763, 349526, 699050, 1398101, 2796202, 5592406, 11184811, 22369622, 44739242, 89478485, 178956970, 357913942, 715827883 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Motivation: consideration of the "hats" problem (which boils down to normal hamming covering codes) in the case when the people are indistinguishable or unlabeled.

Contribution from Paul Curtz, May 26 2011: (Start)

If we add a(0)=1 in front and build the table of a(n) and iterated differences in further rows we get:

1,    1,  2,  2,  5, 10,

0,    1,  0,  3,  5, 12,

1,   -1,  3,  2,  7,  9,

-2,   4, -1,  5,  2, 13,

6,   -5,  6, -3, 11,  6

-11, 11, -9, 14, -5, 21.

The first column is the inverse binomial transform, which is 1,0 followed by (-1)^n*A083322(n-1), n>=2.

The main diagonal in the table above is A001045, the adjacent upper diagonals are A078008, A048573 and A062092. (End)

LINKS

Table of n, a(n) for n=1..31.

Index to sequences with linear recurrences with constant coefficients, signature (2,0,-1,2).

FORMULA

If (n mod 6 = 5) then sum(binomial(n, 3*i+1), i=0..n/3); elif (n mod 6 = 2) then sum(binomial(n, 3*i), i=0..n/3)+1; else sum(binomial(n, 3*i), i=0..n/3); fi;

G.f.: x*(2*x^3-2*x^2+1)/( (1-2*x)*(1+x)*(1-x+x^2) ).

a(n)=2*a(n-1)-a(n-3)+2*a(n-4).

From Paul Curtz, May 26 2011: (Start)

a(n+1) - 2*a(n) has period length 6: repeat 0, -2, 1, 0, 2, -1 (see A080425).

a(n) - A083322(n-1) = A010892(n-1) has period length 6.

a(n) + a(n+3) = 3*2^n = A007283(n).

a(n+6)-a(n) = 21*2^n = A175805(n).

a(n) - A131708(n) = -A131531(n).  (End)

MAPLE

hatwork := proc(n, i, covered) local val, val2; options remember;

# computes the minimum cover of the i-bit through n-bit words.

# if covered is true the i-bit words are already covered (by the (i-1)-bit words)

if (i>n or (i = n and covered)) then 0; elif (i = n and not covered) then 1; else

# one choice is to include the i-bit words in the cover

val := hatwork(n, i+1, true) + binomial(n, i);

# the other choice is not to include the i-bit words in the cover

if (covered) then val2 := hatwork (n, i+1, false); if (val2 < val) then val := val2; fi; else

# if the i-bit words were not covered by (i-1), they must be covered by the (i+1)-bit words

if (i <= n) then val2 := hatwork (n, i+2, true) + binomial(n, i+1); if (val2 < val) then val := val2; fi; fi; fi; val; fi; end proc;

A081374 := proc (n) hatwork(n, 0, false); end proc;

CROSSREFS

Cf. A083322.

Sequence in context: A246864 A247354 A075125 * A243338 A245306 A117400

Adjacent sequences:  A081371 A081372 A081373 * A081375 A081376 A081377

KEYWORD

nonn

AUTHOR

David Applegate, Aug 22 2003

STATUS

approved

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Last modified September 23 04:19 EDT 2014. Contains 247088 sequences.