A230368 A strong divisibility sequence associated with the algebraic integer 1 + i.

1, 1, 1, 5, 1, 1, 1, 15, 1, 1, 1, 65, 1, 1, 1, 255, 1, 1, 1, 1025, 1, 1, 1, 4095, 1, 1, 1, 16385, 1, 1, 1, 65535, 1, 1, 1, 262145, 1, 1, 1, 1048575, 1, 1, 1, 4194305, 1, 1, 1, 16777215, 1, 1, 1, 67108865, 1, 1, 1, 268435455, 1, 1, 1, 1073741825

Offset

1,4

Comments

Let alpha be an algebraic integer and define a sequence of integers a(n) by the condition a(n) = max { integer d : alpha^n == 1 (mod d)}. Silverman shows that a(n) is a strong divisibility sequence, that is gcd(a(n), a(m)) = a(gcd(n, m)) for all n and m in N; in particular, if n divides m then a(n) divides a(m). For the present sequence we take alpha = 1 + i. For other examples see A230369, A235450 and (conjecturally) A082630.

Links

Table of n, a(n) for n=1..60.

J. H. Silverman, Divisibility sequences and powers of algebraic integers, Documenta Mathematica, Extra Volume: John H. Coates' Sixtieth Birthday (2006) 711-727

Index entries for linear recurrences with constant coefficients, signature (0,0,0,4,0,0,0,1,0,0,0,-4).

Formula

a(4*n) = |(-4)^n - 1| otherwise a(n) = 1.

a(4*n) = 5*A015521(n).

O.g.f.: 1/(1 - 4*x^4) - 1/(1 + x^4) + 1/(1 - x) - 1/(1 - x^4) = x*(-1 -x -x^2 -5*x^3 +3*x^4 +3*x^5 +3*x^6 +5*x^7 +4*x^8 +4*x^9 +4*x^10) / ( (1-x) *(1+x) *(2*x^2+1) *(2*x^2-1) *(x^2+1) *(x^4+1) ).

Recurrence equation: a(n) = 4*a(n-4) + a(n-8) - 4*a(n-12).

Maple

seq( gcd( 1/2*((1 - I)^n + (1 + I)^n - 2), I/2*((1 + I)^n - (1 - I )^n ) ), n = 1..80);

Crossrefs

Cf. A247281, A014985, A015521, A082630, A230369, A235450.

Sequence in context: A183097 A353958 A285486 * A256690 A181985 A304320

Adjacent sequences: A230365 A230366 A230367 * A230369 A230370 A230371

Keyword

nonn,easy

Author

Peter Bala, Jan 10 2014

Status

approved