A055495 -id:A055495 - cp's OEIS frontend

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

W. Edwin Clark, Aug 17 2003

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.

			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.

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

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A055495.

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 *)

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

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)

More terms from David Wasserman, Feb 16 2005

A257542 Square-sum pairs: Numbers n such that 0,1, ..., 2n-1 can be partitioned into n pairs, where each pair adds up to a perfect square.

Original entry on oeis.org

1, 4, 5, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Brian Hopkins, Apr 28 2015

Kilkelly uses induction to prove that all integers greater than 20 are in the sequence after using various methods on smaller cases.
The positive integers except 2, 3, and 6.
The positive integers except the strong divisors of 6. - Omar E. Pol, Apr 30 2015

			For n = 4: (0, 1), (2, 7), (3, 6), (4, 5)
For n = 7: (0, 9), (1, 8), (2, 7), (3, 13), (4, 12), (5, 11), (6, 10)

T. Kilkelly, The ARML Power Contest, American Mathematical Society, 2015, chapter 11.

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

Cf. A253472, A252897.

Essentially the same as A055495.

is(n)=n>6 || n==1 || n==4 || n==5 \\ Charles R Greathouse IV, Apr 30 2015

From Chai Wah Wu, Aug 13 2020: (Start)
a(n) = 2*a(n-1) - a(n-2) for n > 5.
G.f.: x*(-x^4 + x^3 - 2*x^2 + 2*x + 1)/(x - 1)^2. (End)

cp's OEIS Frontend

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).

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