A239748 -id:A239748 - cp's OEIS frontend

A238596 Number of distinct sequences defined by the upper left value in powers of n X n (0,1) matrices.

2, 6, 50, 1140, 86052

Offset: 1

Jay Anderson, Mar 01 2014

A sequence can be defined by powers of a matrix with only 0 and 1 values. For instance, the upper left value in the matrix M^n where M=[0 1; 1 1] is the Fibonacci sequence.

Also, the number of distinct sequences defined by all element values in powers of n X n (0, 1) matrices (see A239748) that start with 1. - Christopher Hunt Gribble, May 12 2014

			a(2) = 6 since there are 6 distinct sequences for 2 X 2 (0,1) matrices:
[0 0; 0 0] => 0 0 0 0 0 ...
[1 0; 0 0] => 1 1 1 1 1 ...
[0 1; 0 0] => 0 0 0 0 0 ...
[1 1; 0 0] => 1 1 1 1 1 ...
[0 0; 1 0] => 0 0 0 0 0 ...
[1 0; 1 0] => 1 1 1 1 1 ...
[0 1; 1 0] => 0 1 0 1 0 ...
[1 1; 1 0] => 1 2 3 5 8 ...
[0 0; 0 1] => 0 0 0 0 0 ...
[1 0; 0 1] => 1 1 1 1 1 ...
[0 1; 0 1] => 0 0 0 0 0 ...
[1 1; 0 1] => 1 1 1 1 1 ...
[0 0; 1 1] => 0 0 0 0 0 ...
[1 0; 1 1] => 1 1 1 1 1 ...
[0 1; 1 1] => 0 1 1 2 3 ...
[1 1; 1 1] => 1 2 4 8 16 ...

Cf. A239748.

A240607 a(n) = 2*a(n-2) + a(n-3) + a(n-4) for n>=4, a(n) = binomial(n,3) for n<4.

Original entry on oeis.org

0, 0, 0, 1, 0, 2, 1, 5, 4, 13, 14, 35, 45, 97, 139, 274, 420, 784, 1253, 2262, 3710, 6561, 10935, 19094, 32141, 55684, 94311, 162603, 276447, 475201, 809808, 1389452, 2371264, 4063913, 6941788, 11888542, 20318753, 34782785, 59467836, 101772865, 174037210

Offset: 0

Christopher Hunt Gribble, Apr 09 2014

a(n) = term (4,1) in the 4 X 4 matrix [0,1,1,1; 1,0,1,0; 0,1,0,0; 0,0,1,0]^n. There are 96 ways to define the sequence as an element of the n-th power of a 4 X 4 {0,1}-matrix. These are listed in the second Gribble link.

Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 50 terms from Christopher Hunt Gribble)
Christopher Hunt Gribble, Matrix elements
Index entries for linear recurrences with constant coefficients, signature (0,2,1,1).

Cf. A239748.

a:= proc(n) option remember; `if`(n<4, binomial(n, 3),
       2*a(n-2) +a(n-3) +a(n-4))
    end:
seq(a(n), n=0..50);
# second Maple program using the {0,1}-matrix:
a:= n-> (<<0|1|1|1>, <1|0|1|0>, <0|1|0|0>, <0|0|1|0>>^n)[4, 1]:
seq(a(n), n=0..50);  # Alois P. Heinz, Apr 26 2014

LinearRecurrence[{0,2,1,1},{0,0,0,1,0,2,1,5,4},50] (* Harvey P. Dale, Jul 01 2015 *)

concat([0,0,0], Vec(-x^3/(x^4+x^3+2*x^2-1) + O(x^100))) \\ Colin Barker, Apr 20 2014

G.f.: -x^3 / (x^4+x^3+2*x^2-1). - Colin Barker, Apr 20 2014

More terms from Colin Barker, Apr 20 2014

cp's OEIS Frontend

A238596 Number of distinct sequences defined by the upper left value in powers of n X n (0,1) matrices.

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A240607 a(n) = 2*a(n-2) + a(n-3) + a(n-4) for n>=4, a(n) = binomial(n,3) for n<4.

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