A070367 -id:A070367 - cp's OEIS frontend

A201910 Irregular triangle of 5^k mod prime(n).

1, 1, 2, 0, 1, 5, 4, 6, 2, 3, 1, 5, 3, 4, 9, 1, 5, 12, 8, 1, 5, 8, 6, 13, 14, 2, 10, 16, 12, 9, 11, 4, 3, 15, 7, 1, 5, 6, 11, 17, 9, 7, 16, 4, 1, 5, 2, 10, 4, 20, 8, 17, 16, 11, 9, 22, 18, 21, 13, 19, 3, 15, 6, 7, 12, 14, 1, 5, 25, 9, 16, 22, 23, 28, 24, 4

Offset: 1

T. D. Noe, Dec 07 2011

Except for the third row, the first term of each row is 1. Many sequences are in this one: starting at A036121 (mod 23) and A070365 (mod 7).

			The first 9 rows are:
1
1, 2
0
1, 5, 4, 6, 2, 3
1, 5, 3, 4, 9
1, 5, 12, 8
1, 5, 8, 6, 13, 14, 2, 10, 16, 12, 9, 11, 4, 3, 15, 7
1, 5, 6, 11, 17, 9, 7, 16, 4
1, 5, 2, 10, 4, 20, 8, 17, 16, 11, 9, 22, 18, 21, 13, 19, 3, 15, 6, 7, 12, 14

T. D. Noe, Rows n = 1..60, flattened

Cf. A201908 (2^k), A201909 (3^k), A201911 (7^k).

Cf. A070365 (7), A070367 (11), A070368 (13), A070371 (17), A070373 (19), A036121 (23), A070379 (29), A070384 (37), A070387 (41), A070389 (43), A036127 (47), A036133 (73), A036137 (97), A036139 (103), A036149 (157), A036151 (167), A036156 (193).

P:=Filtered([1..350],IsPrime);;
R:=List([1..Length(P)],n->OrderMod(5,P[n]));;
Flat(Concatenation([1,1,2,0],List([3..10],n->List([0..R[n]-1],k->PowerMod(5,k,P[n]))))); # Muniru A Asiru, Feb 02 2019

nn = 10; p = 5; t = p^Range[0,Prime[nn]]; Flatten[Table[If[Mod[n, p] == 0, {0}, tm = Mod[t, n]; len = Position[tm, 1, 1, 2][[-1,1]]; Take[tm, len-1]], {n, Prime[Range[nn]]}]]

A010691 Period 2: repeat (1,10).

Original entry on oeis.org

1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10, 1, 10

Offset: 0

N. J. A. Sloane

Regular continued fraction of (5+sqrt 35)/10. - R. J. Mathar, Nov 21 2011

Sequence is an infinite palindrome in two ways (numbers and English names): ONE, TEN, ONE, TEN, ONE, TEN, ONE, ... . - Eric Angelini, Sep 16 2023

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,1).

Cf. A036117, A070341, A168429, A070367, A070392, A070404, A048271, A187466.

[10^n mod 11: n in [0..80]]; // Vincenzo Librandi, Aug 24 2011

g:=(1+10*z)/((1-z^2)): gser:=series(g, z=0, 66): seq((coeff(gser, z, n)), n=0..65); # Zerinvary Lajos, Feb 25 2009

PadRight[{},100,{1,10}] (* Harvey P. Dale, Aug 27 2013 *)

a(n) = -9/2*(-1)^n + 11/2.
G.f.: (1+10*z)/(1-z^2). - Zerinvary Lajos, Feb 25 2009
a(n) = 10^n mod 11. - M. F. Hasler, Mar 10 2011
From Nicolas Bělohoubek, Nov 11 2021: (Start)
a(n) = 10/a(n-1). See also A010695.
a(n) = 11 - a(n-1). See also A010712. (End)

A271378 a(n) = 5^n mod 31.

Original entry on oeis.org

1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25, 1, 5, 25

Offset: 0

Vincenzo Librandi, Apr 06 2016

Period 3: repeat [1, 5, 25].

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. similar sequences of the type 5^n mod p, where p is a prime: A070365 (p=7), A070367 (p=11), A070368 (p=13), A070371 (p=17), A070373 (p=19), A036121 (p=23), A070379 (p=29), this sequence (p=31), A070384 (p=37), A070387 (p=41), A070389 (p=43), A036127 (p=47), A036133 (p=73), A036137 (p=97), A271379 (p=101), A036139 (p=103), A036149 (p=157), A271380 (p=163) A036151 (p=167), A036156 (p=193).

Magma
```
[Modexp(5, n, 31): n in [0..100]];
```

&cat [[1,5,25]^^30]; // Bruno Berselli, Apr 07 2016

seq(op([1, 5, 25]), n=0..50); # Wesley Ivan Hurt, Jun 30 2016

Mathematica
```
PowerMod[5, Range[0, 100], 31]
```

x='x+O('x^99); Vec((1+5*x+25*x^2)/(1-x^3)) \\ Altug Alkan, Apr 06 2016

G.f.: (1+5*x+25*x^2)/(1-x^3).
a(n) = a(n-3) for n>2.
a(n) = 5^(n mod 3).
a(n) = (31 - 28*cos(2*n*Pi/3) - 20*sqrt(3)*sin(2*n*Pi/3))/3. - Wesley Ivan Hurt, Jun 30 2016

Edited by Bruno Berselli, Apr 07 2016

A187466 a(n) = 9^n mod 11.

Original entry on oeis.org

1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1, 9, 4, 3, 5, 1

Offset: 0

M. F. Hasler, Mar 10 2011

Period 5: repeat [1, 9, 4, 3, 5].

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,1).

Cf. A010691, A036117, A048271, A070341, A070367, A070392, A070404, A168429.

A187466:=n->9^n mod 11: seq(A187466(n), n=0..100); # Wesley Ivan Hurt, Jun 11 2016

PowerMod[9, Range[0, 100], 11] (* Wesley Ivan Hurt, Jun 11 2016 *)

a(n)=lift(Mod(9,11)^n) \\ Charles R Greathouse IV, Mar 22 2016

G.f.: (5*x^4 + 3*x^3 + 4*x^2 + 9*x + 1)/(1 - x^5). - Chai Wah Wu, Jun 04 2016

a(n) = a(n-5) for n>4. - Wesley Ivan Hurt, Jun 11 2016

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A201910 Irregular triangle of 5^k mod prime(n).

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A010691 Period 2: repeat (1,10).

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A271378 a(n) = 5^n mod 31.

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A187466 a(n) = 9^n mod 11.

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