A085960 Size of the largest code of length 4 and minimum distance 3 over an alphabet of size n. This is usually denoted by A_{n}(4,3).

1, 2, 9, 16, 25, 34, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961, 1024, 1089, 1156, 1225, 1296, 1369, 1444, 1521, 1600, 1681, 1764, 1849, 1936, 2025, 2116, 2209, 2304, 2401, 2500

Offset

1,2

Comments

For n not 2 and not 6, a code C of size n^2 is given by two orthogonal Latin squares A and B of order n by C = {(i,j,A(i,j),B(i,j)): i,j in {1..n}}. Two orthogonal Latin squares of order n exist if and only if n is not 2 and not 6. See A055495.

References

Raymond Hill, "A First Course in Coding Theory", Clarendon Press, Oxford, 1986 (see chapter 10, Theorem 10.16)

Links

Colin Barker, Table of n, a(n) for n = 1..1000

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

Formula

a(n) = 2 if n = 2, a(n) = 34 if n = 6, otherwise a(n) = n^2.

From Colin Barker, Nov 04 2015: (Start)

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>9.

G.f.: -x*(2*x^8-6*x^7+6*x^6-2*x^5+2*x^4-6*x^3+6*x^2-x+1) / (x-1)^3.

(End)

Example

a(2) = 2 since the code C={0000,1110} has minimum distance 3 over the alphabet {0,1} and there is no such code with more codewords.

Mathematica

Table[n^2 - 2 (Boole[n == 2] + Boole[n == 6]), {n, 50}] (* Wesley Ivan Hurt, Nov 04 2015 *)
LinearRecurrence[{3, -3, 1}, {1, 2, 9, 16, 25, 34, 49, 64, 81}, 50] (* Harvey P. Dale, Apr 18 2019 *)

Prog

(PARI) Vec(-x*(2*x^8-6*x^7+6*x^6-2*x^5+2*x^4-6*x^3+6*x^2-x+1)/(x-1)^3 + O(x^100)) \\ Colin Barker, Nov 04 2015

Crossrefs

Cf. A055495.

Sequence in context: A131189 A288484 A011193 * A051386 A003325 A338667

Adjacent sequences: A085957 A085958 A085959 * A085961 A085962 A085963

Keyword

easy,nonn

Author

W. Edwin Clark, Aug 17 2003

Extensions

More terms from David Wasserman, Feb 16 2005

Status

approved