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A128924 T(n,m) is the number of m's in the fundamental period of Fibonacci numbers mod n. 9

%I

%S 1,1,2,2,3,3,1,3,1,1,4,4,4,4,4,2,6,3,4,3,6,2,4,2,1,1,2,4,2,3,2,1,0,3,

%T 0,1,2,5,2,2,2,2,2,2,5,4,8,4,8,4,8,4,8,4,8,1,3,2,1,0,1,0,0,1,0,1,2,5,

%U 2,2,1,5,0,1,1,2,2,1,4,4,2,2,0,4,0,0,4,0,2,2,4,2,8,2,2,1,4,4,4,4,4,1,2,2,8

%N T(n,m) is the number of m's in the fundamental period of Fibonacci numbers mod n.

%C T(n,m) is the triangle read by rows, 0<=m<n. First column is A001176. Row sums are A001175.

%C A118965 and A066853 give numbers of zeros and nonzeros in n-th row, respectively. - _Reinhard Zumkeller_, Jan 16 2014

%C T(n,n) = A235715(n). - _Reinhard Zumkeller_, Jan 17 2014

%H Reinhard Zumkeller, <a href="/A128924/b128924.txt">Rows n = 1..125 of triangle, flattened</a>

%H G. Darvasi and St. Eckmann, <a href="https://eudml.org/doc/141577">Zur Verteilung der Reste der Fibonacci-Folge modulo 5c</a>, Elemente der Mathematik 50 (1995) pp. 76-80.

%e {F(k) mod 4} has fundamental period (0,1,1,2,3,1), see A079343, with

%e T(4,0)=1 zero, T(4,1)=3 ones, T(4,2)=1 two's, T(4,3)=1 three's. The triangle starts

%e 1,

%e 1, 2,

%e 2, 3, 3,

%e 1, 3, 1, 1,

%e 4, 4, 4, 4, 4,

%e 2, 6, 3, 4, 3, 6,

%e 2, 4, 2, 1, 1, 2, 4,

%e 2, 3, 2, 1, 0, 3, 0, 1,

%e 2, 5, 2, 2, 2, 2, 2, 2, 5,

%e 4, 8, 4, 8, 4, 8, 4, 8, 4, 8,

%e 1, 3, 2, 1, 0, 1, 0, 0, 1, 0, 1,

%e 2, 5, 2, 2, 1, 5, 0, 1, 1, 2, 2, 1,

%e 4, 4, 2, 2, 0, 4, 0, 0, 4, 0, 2, 2, 4,

%e 2, 8, 2, 2, 1, 4, 4, 4, 4, 4, 1, 2, 2, 8,

%e 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3,

%e 2, 3, 4, 1, 0, 3, 0, 1, 2, 3, 0, 1, 0, 3, 0, 1,

%e 4, 4, 2, 2, 4, 2, 0, 0, 2, 2, 0, 0, 2, 4, 2, 2, 4,

%p A128924 := proc(m,h)

%p local resul,k,M ;

%p resul :=0 ;

%p for k from 0 to A001175(m)-1 do

%p M := combinat[fibonacci](k) mod m ;

%p if M = h then

%p resul := resul+1 ;

%p end if ;

%p end do;

%p resul ;

%p end proc:

%p seq(seq(A128924(m,h),h=0..m-1),m=1..17) ;

%t A001175[1] = 1; A001175[n_] := For[k = 1, True, k++, If[Mod[Fibonacci[k], n] == 0 && Mod[Fibonacci[k+1], n] == 1, Return[k]]]; T[m_, h_] := Module[{resul, k, M}, resul = 0; For[k = 0, k <= A001175[m]-1, k++, M = Mod[Fibonacci[k], m]; If[ M == h, resul++]]; Return[resul]]; Table[T[m, h], {m, 1, 17}, {h, 0, m-1}] // Flatten (* _Jean-Fran├žois Alcover_, Feb 11 2015, after Maple code *)

%o (Haskell)

%o import Data.List (group, sort)

%o a128924 n k = a128924_tabl !! (n-1) !! (k-1)

%o a128924_tabl = map a128924_row [1..]

%o a128924_row 1 = [1]

%o a128924_row n = f [0..n-1] $ group $ sort $ g 1 ps where

%o f [] _ = []

%o f (v:vs) wss'@(ws:wss) | head ws == v = length ws : f vs wss

%o | otherwise = 0 : f vs wss'

%o g 0 (1 : xs) = []

%o g _ (x : xs) = x : g x xs

%o ps = 1 : 1 : zipWith (\u v -> (u + v) `mod` n) (tail ps) ps

%o -- _Reinhard Zumkeller_, Jan 16 2014

%Y Cf. A053029, A053030, A053031.

%K nonn,tabl

%O 1,3

%A _R. J. Mathar_, Apr 25 2007

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Last modified April 17 09:29 EDT 2021. Contains 343064 sequences. (Running on oeis4.)