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A128924 T(n,m) is the number of m's in the fundamental period of Fibonacci numbers mod n. 8
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, 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, 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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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

A118965 and A066853 give numbers of zeros and nonzeros in n-th row, respectively. - Reinhard Zumkeller, Jan 16 2014

T(n,n) = A235715(n). - Reinhard Zumkeller, Jan 17 2014

LINKS

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened

G. Darvasi and St. Eckmann, Zur Verteilung der Reste der Fibonacci-Folge modulo 5c, Elemente der Mathematik 50 (1995) pp. 76-80.

EXAMPLE

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

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

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, 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, 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,

2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3, 2, 3, 3,

2, 3, 4, 1, 0, 3, 0, 1, 2, 3, 0, 1, 0, 3, 0, 1,

4, 4, 2, 2, 4, 2, 0, 0, 2, 2, 0, 0, 2, 4, 2, 2, 4,

MAPLE

A128924 := proc(m, h)

    local resul, k, M ;

    resul :=0 ;

    for k from 0 to A001175(m)-1 do

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

        if M = h then

            resul := resul+1 ;

        end if ;

    end do;

    resul ;

end proc:

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

MATHEMATICA

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 *)

PROG

(Haskell)

import Data.List (group, sort)

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

a128924_tabl = map a128924_row [1..]

a128924_row 1 = [1]

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

   f []     _                            = []

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

                          | otherwise    = 0 : f vs wss'

   g 0 (1 : xs) = []

   g _ (x : xs) = x : g x xs

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

-- Reinhard Zumkeller, Jan 16 2014

CROSSREFS

Cf. A053029, A053030, A053031.

Sequence in context: A105899 A071434 A227314 * A239957 A230040 A242361

Adjacent sequences:  A128921 A128922 A128923 * A128925 A128926 A128927

KEYWORD

nonn,tabl

AUTHOR

R. J. Mathar, Apr 25 2007

STATUS

approved

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Last modified August 23 04:06 EDT 2017. Contains 290958 sequences.