0,2

[Horadam], a 2 X 2 Hadamard matrix = [N, N; N, -N]. Conjecture: Let such 2 X 2 matrices = H. Exracting upper left terms from the matrices H^n, n=0,1,2; we obtain sequences of the form 1, H,...; then odd indexed k-th terms = N * previous term, and even indexed k-th terms = 2N * previous term. Example. N=5, H = [5,5; 5,-5]. The prescribed operation generates (1, 5, 50, 250, 2500, 12500,...). Even powers of H = k(n)*I, where k(n) = k-th term in the sequence and I = identity matrix. Odd powers of H = another Hadamard matrix [k(n),k(n); k(n), -k(n)]. Example using A130505: N=3, H = [3,3; 3,-3]. Then H^4 = [324,0; 0, 324] = a(4)*I since a(4) = 324. But H^3 = [54,54; 54,-54], another Hadamard matrix.

Horadam, K.J., "Hadamard Matrices and Their Applications", Princeton University Press, 2006.

a(0) = 1; a(n), n>0 = 3*a(n-1) if n is odd. a(n), n even = 6 * a(n-1). Upper left term in M^n where M = the 2 X 2 Hadamard matrix [3, 3; 3, -3].

a(3) = 54 = 3 * a(2) = 3 * 18.

a(4) = 324 = 6 * a(3) = 6 * 54.

Sequence in context: A064043 A085789 A027334 this_sequence A027289 A061317 A139362

Adjacent sequences: A130502 A130503 A130504 this_sequence A130506 A130507 A130508

nonn

Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 01 2007