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A154754 Ratio of the period and the reduced period of the Fibonacci 3-step sequence A000073 mod prime(n).

%I A154754 #12 Jul 24 2025 09:36:17
%S A154754 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,
%T A154754 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,
%U A154754 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
%N A154754 Ratio of the period and the reduced period of the Fibonacci 3-step sequence A000073 mod prime(n).
%C A154754 See A046737 for more information about the reduced period.
%C A154754 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 1's as 3's.
%F A154754 a(n) = A106302(n) / A154753(n).
%F A154754 a(n) = A386236(prime(n)), where prime(n) is the n-th prime.
%e A154754 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.
%t A154754 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}]
%Y A154754 See the comments for the relationships with A116515, A154755, A154756.
%Y A154754 See the formula section for the relationships with A106302, A154753, A386236.
%Y A154754 Cf. A000073.
%Y A154754 For the periods modulo all positive integers see A046737, A046738.
%K A154754 nonn
%O A154754 1,4
%A A154754 _T. D. Noe_, Jan 15 2009