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A154754
Ratio of the period and the reduced period of the Fibonacci 3-step sequence A000073 mod prime(n).
3
1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 3, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 1, 3, 1, 3, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 3
OFFSET
1,4
COMMENTS
For the Fibonacci 3-step (tribonacci) sequence, only 1 and 3 appear. A116515 is the analogous sequence for Fibonacci numbers. Let the terms in the reduced period be denoted by R. When the ratio is 3, the full period can be written as R,aR,bR, where a and b are multipliers that are the two solutions of the equation x^2+x+1 = 0 (mod p). What order do the solutions appear as a and b? See A154755 and A154756 for the primes that produce ratios of 1 and 3, respectively. Observe that there are approximately three times as many 1s as 3s.
FORMULA
a(n) = A106302(n) / A154753(n)
EXAMPLE
The tribonacci sequence (starting with 1) mod 7 is 1,1,2,4,0,6,3,2,4, 2,1,0,3,4,0,0,4,4,1,2,0,3,5,1,2,1,4,0,5,2,0,0,2,2,4,1,0,5,6,4,1,4,2,0, 6,1,0,0, which has 3 pairs of 0-0 terms. Hence a(4)=3.
MATHEMATICA
Table[p=Prime[i]; a={1, 0, 0}; a0=a; k=0; zeros=0; While[k++; s=Mod[Plus@@a, p]; a=RotateLeft[a]; a[[ -1]]=s; If[Rest[a]=={0, 0}, zeros++ ]; a!=a0]; zeros, {i, 200}]
CROSSREFS
Sequence in context: A138291 A201681 A062174 * A102368 A374146 A277109
KEYWORD
nonn
AUTHOR
T. D. Noe, Jan 15 2009
STATUS
approved