A344309 a(n) is the number of eigensequences (including eigenvalues, where they exist) of the Fibonacci recurrence in Zp where p is prime(n).
0, 0, 1, 0, 2, 1, 1, 2, 0, 3, 2, 1, 3, 0, 2, 1, 2, 5, 0, 2, 1, 2, 0, 9, 1, 3, 0, 2, 5, 5, 0, 2, 1, 4, 5, 4, 1, 0, 0, 1, 2, 3, 2, 1, 1, 10, 6, 0, 0, 3, 17, 2, 3, 2, 1, 2, 5, 2, 1, 11, 0, 1, 6, 2, 1, 1, 4, 1, 2, 3, 5, 2, 0, 1, 2, 0, 5, 1, 5, 3, 2, 21, 2, 1, 2, 0
Offset: 1
Keywords
Links
- M. Aoki, Y. Sakai, On Equivalence Classes of Generalized Fibonacci Sequences, JIS vol 19 (2016) # 16.2.6
- H. Sedaghat, Zero-Avoiding Solutions of the Fibonacci Recurrence Modulo A Prime, Fibonacci Quart. 52 (2014), no. 1, 39-45. See p. 44.
Programs
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Maple
A344309 := proc(i) local p,F ; p := ithprime(i) ; for n from 1 do F := combinat[fibonacci](n) ; if modp(F,p) =0 then if modp(p,5) in {2,3} then return (p+1)/n-1 ; elif modp(p,5) in {1,4} then return (p-1)/n+1 ; else return 1 ; end if; end if; end do: end proc; seq(A344309(n),n=1..55) ; # R. J. Mathar, Feb 27 2023
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PARI
ep(n) = if(n==3, 5, my(p=prime(n)); fordiv(p^2-1, d, if(fibonacci(d)%p==0, return(d)))); \\ A001602 a(n) = {my(p=prime(n), mp = p % 5); my(x=if ((mp==0), 4, if ((mp==1) || (mp==4), (p-1)^2/ep(n) + p - 1, (p^2-1)/ep(n) + 1 - p))); x/(p-1);}
Formula
a(n) = E(p)/(p-1) where p is prime(n) and E(p) = 4 if p=5; (p-1)^2/ep(n) + p - 1 if p==1 or 4 (mod 5); (p^2-1)/ep(n) + 1 - p if p==2 or 3 (mod 5), where ep(n) = A001602(n).
Comments