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A128924 T(n,m) is the number of m's in the fundamental period of Fibonacci numbers mod n. 9
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.
A118965 and A066853 give numbers of zeros and nonzeros in n-th row, respectively. - Reinhard Zumkeller, Jan 16 2014
LINKS
G. Darvasi and St. Eckmann, Zur Verteilung der Reste der Fibonacci-Folge modulo 5c, Elemente der Mathematik 50 (1995) pp. 76-80.
FORMULA
T(n,n) = A235715(n). - Reinhard Zumkeller, Jan 17 2014
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-François 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, A001175 (row sums), A001176 (1st column).
Sequence in context: A366549 A227314 A356321 * A364569 A239957 A230040
KEYWORD
nonn,tabl
AUTHOR
R. J. Mathar, Apr 25 2007
STATUS
approved

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Last modified March 28 12:59 EDT 2024. Contains 371254 sequences. (Running on oeis4.)