A175182 -id:A175182 - cp's OEIS frontend

A322907 Entry points for the 3-Fibonacci numbers A006190.

1, 3, 2, 6, 3, 6, 8, 6, 6, 3, 4, 6, 13, 24, 6, 12, 8, 6, 20, 6, 8, 12, 22, 6, 15, 39, 18, 24, 7, 6, 32, 24, 4, 24, 24, 6, 19, 60, 26, 6, 7, 24, 42, 12, 6, 66, 48, 12, 56, 15, 8, 78, 26, 18, 12, 24, 20, 21, 12, 6, 30, 96, 24, 48, 39, 12, 68, 24, 22, 24, 72, 6

Offset: 1

Jianing Song, Jan 05 2019

nonn

a(n) is the smallest k > 0 such that n divides A006190(k).
a(n) is also called the rank of A006190(n) modulo n.
For primes p == 1, 9, 17, 25, 29, 49 (mod 52), a(p) divides (p - 1)/2.
For primes p == 3, 23, 27, 35, 43, 51 (mod 52), a(p) divides p - 1, but a(p) does not divide (p - 1)/2.
For primes p == 5, 21, 33, 37, 41, 45 (mod 52), a(p) divides (p + 1)/2.
For primes p == 7, 11, 15, 19, 31, 47 (mod 52), a(p) divides p + 1, but a(p) does not divide (p + 1)/2.
a(n) <= (12/7)*n for all n, where the equality holds if and only if n = 2*7^e, e >= 1.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..5000 from Jianing Song)

Let {x(n)} be a sequence defined by x(0) = 0, x(1) = 1, x(n+2) = k*x(n+1) + x(n). Then the periods, ranks and the ratios of the periods to the ranks modulo a given integer n are given by:
k = 1: A001175 (periods), A001177 (ranks), A001176 (ratios).
k = 2: A175181 (periods), A214028 (ranks), A214027 (ratios).
k = 3: A175182 (periods), this sequence (ranks), A322906 (ratios).
Cf. A006190.

A006190(m) = ([3, 1; 1, 0]^m)[2, 1]
a(n) = my(i=1); while(A006190(i)%n!=0, i++); i

a(m*n) = a(m)*a(n) if gcd(m, n) = 1.
For odd primes p, a(p^e) = p^(e-1)*a(p) if p^2 does not divide a(p). Any counterexample would be a 3-Wall-Sun-Sun prime.
a(2^e) = 3 if e = 1, 6 if e = 2 and 3*2^(e-2) if e >= 3. a(13^e) = 13^e, e >= 1.

A175183 Pisano period of the 4-Fibonacci numbers A001076.

Original entry on oeis.org

1, 2, 8, 2, 20, 8, 16, 4, 8, 20, 10, 8, 28, 16, 40, 8, 12, 8, 6, 20, 16, 10, 16, 8, 100, 28, 24, 16, 14, 40, 10, 16, 40, 12, 80, 8, 76, 6, 56, 20, 40, 16, 88, 10, 40, 16, 32, 8, 112, 100, 24, 28, 36, 24, 20, 16, 24, 14, 58, 40, 20, 10, 16, 32, 140, 40, 136, 12, 16, 80, 70, 8, 148, 76

Offset: 1

R. J. Mathar, Mar 01 2010

Period of the sequence defined by reading A001076 modulo n.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Sergio Falcon and Ángel Plaza, k-Fibonacci sequences modulo m, Chaos, Solit. Fractals 41 (2009), 497-504.
Eric Weisstein's World of Mathematics, Pisano period.
Wikipedia, Pisano period.

Cf. A001076, A001175, A175181, A175182, A175184, A175185.

F := proc(k,n) option remember; if n <= 1 then n; else k*procname(k,n-1)+procname(k,n-2) ; end if; end proc:
Pper := proc(k,m) local cha, zer,n,fmodm ; cha := [] ; zer := [] ; for n from 0 do fmodm := F(k,n) mod m ; cha := [op(cha),fmodm] ; if fmodm = 0 then zer := [op(zer),n] ; end if; if nops(zer) = 5 then break; end if; end do ; if [op(1..zer[2],cha) ] = [ op(zer[2]+1..zer[3],cha) ] and [op(1..zer[2],cha)] = [ op(zer[3]+1..zer[4],cha) ] and [op(1..zer[2],cha)] = [ op(zer[4]+1..zer[5],cha) ] then return zer[2] ; elif [op(1..zer[3],cha) ] = [ op(zer[3]+1..zer[5],cha) ] then return zer[3] ; else return zer[5] ; end if; end proc:
k := 4 ; seq( Pper(k,m),m=1..80) ;

Table[s = t = Mod[{0, 1}, n]; cnt=1; While[tmp = Mod[4*t[[2]] + t[[1]], n]; t[[1]] = t[[2]]; t[[2]] = tmp; s!= t, cnt++]; cnt, {n, 100}] (* Vincenzo Librandi, Dec 20 2012, after T. D. Noe *)

A175184 Pisano period of the 5-Fibonacci numbers, A052918 preceded by 0.

Original entry on oeis.org

1, 3, 8, 6, 2, 24, 6, 12, 8, 6, 24, 24, 12, 6, 8, 24, 36, 24, 40, 6, 24, 24, 22, 24, 10, 12, 8, 6, 116, 24, 64, 48, 24, 36, 6, 24, 76, 120, 24, 12, 28, 24, 88, 24, 8, 66, 96, 24, 42, 30, 72, 12, 52, 24, 24, 12, 40, 348, 58, 24, 124, 192, 24, 96, 12, 24, 66, 36, 88, 6, 70, 24, 148

Offset: 1

R. J. Mathar, Mar 01 2010

Period of the sequence defined by reading A052918 modulo n.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Sergio Falcon and Ángel Plaza, k-Fibonacci sequences modulo m, Chaos, Solit. Fractals 41 (2009), 497-504.
Eric Weisstein's World of Mathematics, Pisano period.
Wikipedia, Pisano period.

Cf. A001175, A052918, A175181, A175182, A175183, A175185.

F := proc(k,n) option remember; if n <= 1 then n; else k*procname(k,n-1)+procname(k,n-2) ; end if; end proc:
Pper := proc(k,m) local cha, zer,n,fmodm ; cha := [] ; zer := [] ; for n from 0 do fmodm := F(k,n) mod m ; cha := [op(cha),fmodm] ; if fmodm = 0 then zer := [op(zer),n] ; end if; if nops(zer) = 5 then break; end if; end do ; if [op(1..zer[2],cha) ] = [ op(zer[2]+1..zer[3],cha) ] and [op(1..zer[2],cha)] = [ op(zer[3]+1..zer[4],cha) ] and [op(1..zer[2],cha)] = [ op(zer[4]+1..zer[5],cha) ] then return zer[2] ; elif [op(1..zer[3],cha) ] = [ op(zer[3]+1..zer[5],cha) ] then return zer[3] ; else return zer[5] ; end if; end proc:
k := 5 ; seq( Pper(k,m),m=1..80) ;

Table[s = t = Mod[{0, 1}, n]; cnt = 1; While[tmp = Mod[5*t[[2]] + t[[1]], n]; t[[1]] = t[[2]]; t[[2]] = tmp; s!= t, cnt++]; cnt, {n, 100}] (* Vincenzo Librandi, Dec 20 2012, after T. D. Noe *)

A175289 Pisano period of A002605 modulo n.

Original entry on oeis.org

1, 1, 3, 1, 24, 3, 48, 1, 9, 24, 10, 3, 12, 48, 24, 1, 144, 9, 180, 24, 48, 10, 22, 3, 120, 12, 27, 48, 840, 24, 320, 1, 30, 144, 48, 9, 36, 180, 12, 24, 280, 48, 308, 10, 72, 22, 46, 3, 336, 120, 144, 12, 936, 27, 120, 48, 180, 840, 29, 24, 60, 320, 144, 1, 24, 30, 1122, 144

Offset: 1

R. J. Mathar, Mar 24 2010

nonn

a(79)=6240. [John W. Layman, Aug 10 2010]

			Reading 0, 1, 2, 6, 16, 44, 120, 328, 896, 2448,.. modulo 12 gives 0, 1, 2, 6, 4, 8, 0, 4, 8, 0, 4, 8 ,.. with period length a(n=12)= 3.

Michael De Vlieger, Table of n, a(n) for n = 1..200

Cf. A001175, A046738, A175181, A175182, A175183, A175184, A175185, A175286.

a={1};For[n=2,n<=80,n++,{x={{0,1}}; t={1,1}; While[ !MemberQ[x,t], {xl = x[[ -1]]; AppendTo[x,t]; t={Mod[2*(t[[1]]+xl[[1]]),n], Mod[2*(t[[2]] + xl[[2]]),n]};}]; p = Flatten[Position[x,t]][[1]]; AppendTo[a, Length[x] - p+1];}]; Print[a]; (* John W. Layman, Aug 10 2010 *)

Terms beyond a(28)=48 from John W. Layman, Aug 10 2010

A253247 Pisano period of A006190(n^2) divided by Pisano period of A006190(n).

Original entry on oeis.org

1, 2, 3, 4, 5, 1, 7, 8, 9, 5, 11, 4, 13, 7, 5, 16, 17, 9, 19, 10, 21, 11, 23, 8, 25, 13, 27, 7, 29, 5, 31, 32, 33, 17, 35, 36, 37, 19, 39, 40, 41, 7, 43, 11, 45, 23, 47, 16, 49, 25, 51, 26, 53, 27, 55, 14, 57, 29, 59, 10, 61, 31, 63, 64, 65, 11, 67, 17, 69, 35, 71, 72

Offset: 1

Eric Chen, Apr 09 2015

nonn

For all n, a(n)|n.
Conjecture: a(n) = 1 only for n = 1 and 6. (This conjecture is true if and only if the generalized Wall's conjecture to A006190 is true.)
If there exists any prime p such that A175182(p^2) = A175182(p), then the conjecture fails.
For any prime p, these three statements are equivalent:
(1) A175182(p^2) = A175182(p).
(2) A006190(p-(p|13)) = 3 (mod p^2).
(3) A006497(p) = 1 (mod p^2).
Since A175182(241^2) = A175182(241) = 484, so the prime 241 is a Wall-Sun-Sun prime to A006190 (Lucas (P, Q) = (3, -1)) and no others < 10^8, so the conjecture is true for all primes < 10^8 except 241.
All of Wall's theorems are true for A175182. For example, let P(n) = A175182(n), p and q are primes, then P(pq) = lcm(P(p), P(q)), and for every prime p, P(p)|(p-1) if (p|13) = 1, P(p)|(2p+2) if (p|13) = -1 (P(13) = 52, which if divisible by 13), while (p|13) is the Legendre symbol, and the fixed points of A175182 are 1, 6, and 12*13^k, k>0.

Eric Chen, Table of n, a(n) for n = 1..1000

Cf. A237517, A175182.

F := proc(k, n) option remember; if n <= 1 then n; else k*procname(k, n-1)+procname(k, n-2) ; end if; end proc:
Pper := proc(k, m) local cha, zer, n, fmodm ; cha := [] ; zer := [] ; for n from 0 do fmodm := F(k, n) mod m ; cha := [op(cha), fmodm] ; if fmodm = 0 then zer := [op(zer), n] ; end if; if nops(zer) = 5 then break; end if; end do ; if [op(1..zer[2], cha) ] = [ op(zer[2]+1..zer[3], cha) ] and [op(1..zer[2], cha)] = [ op(zer[3]+1..zer[4], cha) ] and [op(1..zer[2], cha)] = [ op(zer[4]+1..zer[5], cha) ] then return zer[2] ; elif [op(1..zer[3], cha) ] = [ op(zer[3]+1..zer[5], cha) ] then return zer[3] ; else return zer[5] ; end if; end proc:
k := 3 ; seq( Pper(k, m^2) div Pper(k, m), m=1..300) ;

A006190[n_] := Fibonacci[n, 3];
A175182[n_] := Module[{k=1}, While[Mod[A006190[k], n] != 0 || Mod[A006190[k+1]-1, n] != 0, k++]; k];
Table[A175182[n^2] / A175182[n], {n, 72}] (* corrected by Jason Yuen, Jun 28 2025 *)

fibmod(n, m)=((Mod([3, 1; 1, 0], m))^n)[1, 2]
entry_p(p)=my(k=1, c=Mod(1, p), o); while(c, [o, c]=[c, 3*c+o]; k++); k
entry(n)=if(n==1, return(1)); my(f=factor(n), v); v=vector(#f~, i, if(f[i, 1]>1e8 && f[i, 1] != 241, entry_p(f[i, 1]^f[i, 2]), entry_p(f[i, 1])*f[i, 1]^(f[i, 2] - 1))); if(f[1, 1]==2&&f[1, 2]>1, v[1]=3<

a(n) = A175182(n^2) / A175182(n).

A175291 Pisano period of A006130 modulo n.

Original entry on oeis.org

1, 3, 1, 6, 24, 3, 24, 6, 3, 24, 120, 6, 156, 24, 24, 12, 16, 3, 90, 24, 24, 120, 22, 6, 120, 156, 9, 24, 28, 24, 240, 24, 120, 48, 24, 6, 171, 90, 156, 24, 336, 24, 42, 120, 24, 66, 736, 12, 168, 120, 16, 156, 52, 9, 120, 24, 90, 84, 3480, 24, 20, 240, 24, 48, 312, 120, 748, 48, 22, 24, 5040, 6, 888, 171, 120, 90, 120, 156, 39

Offset: 1

R. J. Mathar, Mar 24 2010

nonn

			Reading 0, 1, 1, 4, 7, 19, 40, 97, 217, 508, 1159, 2683, 6160, 14209, ... modulo n=7 gives 0, 1, 1, 4, 0, 5, 5, 6, 0, 4, 4, 2, 0, 6, 6, 3, 0, 2, 2, 1, 0, 3, 3, 5, 0, 1, 1, 4, 0, 5, 5, 6, 0, 4, 4, 2, 0, 6, 6, 3, 0, ... with period a(n=7)=24.

Cf. A001175, A046738, A175181, A175182, A175183, A175184, A175185, A175286.

a(9) corrected by R. J. Mathar, Apr 18 2010

A253246 Pisano period of A006190 to mod prime(n).

Original entry on oeis.org

3, 2, 12, 16, 8, 52, 16, 40, 22, 28, 64, 76, 28, 42, 96, 26, 24, 30, 136, 144, 148, 26, 168, 180, 196, 50, 102, 106, 20, 112, 126, 10, 92, 138, 300, 304, 156, 328, 336, 86, 178, 180, 190, 388, 396, 198, 30, 448, 456, 460, 116, 160, 484, 250, 128, 262, 268, 544, 138, 564

Offset: 1

Eric Chen, Apr 11 2015

nonn

If the generalized Wall's conjecture to A006190 is true, then we can calculate A175182(m) when m is a prime power since for any k>=1 : A175182(prime(n)^k)=a(n)*prime(n)^(k-1). For example: A175182(2^k)=3*2^(k-1)=A007283(k-1).

In fact, the conjecture fails on p=241, and this is the only counterexample below 10^8.

Eric Chen, Table of n, a(n) for n = 1..1000

Cf. A060305, A175182.

Table[s = t = Mod[{0, 1}, Prime[n]]; cnt = 1; While[tmp = Mod[3*t[[2]] + t[[1]], n]; t[[1]] = t[[2]]; t[[2]] = tmp; s!= t, cnt++]; cnt, {n, 100}]

fibmod(n, m)=((Mod([3, 1; 1, 0], m))^n)[1, 2]
entry(p)=my(k=1, c=Mod(1, p), o); while(c, [o, c]=[c, 3*c+o]; k++); ka(n)=entry(prime(n))

a(n) = A175182(A000040(n)).

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A322907 Entry points for the 3-Fibonacci numbers A006190.

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A175183 Pisano period of the 4-Fibonacci numbers A001076.

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A175184 Pisano period of the 5-Fibonacci numbers, A052918 preceded by 0.

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A175289 Pisano period of A002605 modulo n.

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A253247 Pisano period of A006190(n^2) divided by Pisano period of A006190(n).

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A175291 Pisano period of A006130 modulo n.

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A253246 Pisano period of A006190 to mod prime(n).

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