cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A201908 Irregular triangle of 2^k mod (2n-1).

Original entry on oeis.org

0, 1, 2, 1, 2, 4, 3, 1, 2, 4, 1, 2, 4, 8, 7, 5, 1, 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1, 2, 4, 8, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1, 2, 4, 8, 16, 11, 1, 2, 4, 8, 16, 9
Offset: 1

Views

Author

T. D. Noe, Dec 07 2011

Keywords

Comments

The length of the rows is given by A002326. For n > 1, the first term of row n is 1 and the last term is n. Many sequences are in this one: starting at A036117 (mod 11) and A070335 (mod 23).
Row n, for n >= 2, divided elementwise by (2*n-1) gives the cycles of iterations of the doubling function D(x) = 2*x or 2*x-1 if 0 <= x < 1/2 or , 1/2 <= x < 1, respectively, with seed 1/(2*n-1). See the Devaney reference, pp. 25-26. D^[k](x) = frac(2^k/(2*n-1)), for k = 0, 1, ..., A002326(n-1) - 1. E.g., n = 3: 1/5, 2/5, 4/5, 3/5. - Gary W. Adamson and Wolfdieter Lang, Jul 29 2020.

Examples

			The irregular triangle T(n, k) begins:
n\k  0 1 2 3  4  5  6  7 8  9 10 11 12 13 14 15 16 17 ...
---------------------------------------------------------
1:   0
2:   1 2
3:   1 2 4 3
4:   1 2 4
5:   1 2 4 8  7  5
6:   1 2 4 8  5 10  9  7 3  6
7:   1 2 4 8  3  6 12 11 9  5 10  7
8:   1 2 4 8
9:   1 2 4 8 16 15 13  9
10:  1 2 4 8 16 13  7 14 9 18 17 15 11  3  6 12  5 10
... reformatted by _Wolfdieter Lang_, Jul 29 2020.

References

  • Robert L. Devaney, A First Course in Chaotic Dynamical Systems, Addison-Wesley., 1992. pp. 24-25

Links

Crossrefs

Cf. A002326, A201909 (3^k), A201910 (5^k), A201911 (7^k).
Cf. A000034 (3), A070402 (5), A069705 (7), A036117 (11), A036118 (13), A062116 (17), A036120 (19), A070347 (21), A070335 (23), A070336 (25), A070337 (27), A036122 (29), A070338 (33), A070339 (35), A036124 (37), A070340 (39), A070348 (41), A070349 (43), A070350 (45), A070351 (47), A036128 (53), A036129 (59), A036130 (61), A036131 (67), A036135 (83), A036138 (101), A036140 (107), A201920 (125), A036144 (131), A036146 (139), A036147 (149), A036150 (163), A036152 (173), A036153 (179), A036154 (181), A036157 (197), A036159 (211), A036161 (227).

Programs

  • GAP
    R:=List([0..72],n->OrderMod(2,2*n+1));;
Flat(Concatenation([0],List([2..11],n->List([0..R[n]-1],k->PowerMod(2,k,2*n-1))))); # Muniru A Asiru, Feb 02 2019
  • Mathematica
    nn = 30; p = 2; t = p^Range[0, nn]; Flatten[Table[If[IntegerQ[Log[p, n]], {0}, tm = Mod[t, n]; len = Position[tm, 1, 1, 2][[-1,1]]; Take[tm, len-1]], {n, 1, nn, 2}]]

Formula

T(n, k) = 2^k mod (2*n-1), n >= 1, k = 0, 1, ..., A002326(n-1) - 1.
T(n, k) = (2*n-1)*frac(2^k/(2*n-1)), n >= 1, k = 0, 1, ..., A002326(n-1) - 1, with the fractional part frac(x) = x - floor(x). - Wolfdieter Lang, Jul 29 2020

A201912 Irregular triangle of 2^k mod prime(n).

Original entry on oeis.org

0, 1, 2, 1, 2, 4, 3, 1, 2, 4, 1, 2, 4, 8, 5, 10, 9, 7, 3, 6, 1, 2, 4, 8, 3, 6, 12, 11, 9, 5, 10, 7, 1, 2, 4, 8, 16, 15, 13, 9, 1, 2, 4, 8, 16, 13, 7, 14, 9, 18, 17, 15, 11, 3, 6, 12, 5, 10, 1, 2, 4, 8, 16, 9, 18, 13, 3, 6, 12, 1, 2, 4, 8, 16, 3, 6, 12, 24
Offset: 1

Views

Author

T. D. Noe, Dec 17 2011

Keywords

Comments

The row lengths are in A014664. For n > 1, the first term of each row is 1 and the last term is 2*prime(n)-1, which is A006254. Many sequences are in this one.

Examples

			The first 11 rows are:
2:  0;
3:  1, 2;
5:  1, 2, 4, 3;
7:  1, 2, 4;
11: 1, 2, 4, 8,  5, 10,  9,  7,  3,  6;
13: 1, 2, 4, 8,  3,  6, 12, 11,  9,  5, 10,  7;
17: 1, 2, 4, 8, 16, 15, 13,  9;
19: 1, 2, 4, 8, 16, 13,  7, 14,  9, 18, 17, 15, 11,  3,  6, 12,  5, 10;
23: 1, 2, 4, 8, 16,  9, 18, 13,  3,  6, 12;
29: 1, 2, 4, 8, 16,  3,  6, 12, 24, 19,  9, 18,  7, 14, 28, 27, 25, 21, 13, 26, 23, 17, 5, 10, 20, 11, 22, 15;
31: 1, 2, 4, 8, 16;

Links

Crossrefs

Cf. A062117, A201908, A201909, A201910, A201911.
Cf. similar sequences of the type 2^n mod p, where p is a prime: A000034 (p=3), A070402 (p=5), A069705 (p=7), A036117 (p=11), A036118 (p=13), A062116 (p=17), A036120 (p=19), A070335 (p=23), A036122 (p=29), A269266 (p=31), A036124 (p=37), A070348 (p=41), A070349 (p=43), A070351 (p=47), A036128 (p=53), A036129 (p=59), A036130 (p=61), A036131 (p=67), A036135 (p=83), A036138 (p=101), A036140 (p=107), A036144 (p=131), A036146 (p=139), A036147 (p=149), A036150 (p=163), A036152 (p=173), A036153 (p=179), A036154 (p=181), A036157 (p=197), A036159 (p=211), A036161 (p=227).

Programs

  • GAP
    P:=Filtered([1..350],IsPrime);;
R:=List([1..Length(P)],n->OrderMod(2,P[n]));;
Flat(Concatenation([0],List([2..10],n->List([0..R[n]-1],k->PowerMod(2,k,P[n]))))); # Muniru A Asiru, Feb 01 2019
  • Mathematica
    nn = 10; p = 2; 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]]}]]

A269266 a(n) = 2^n mod 31.

Original entry on oeis.org

1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 1
Offset: 0

Views

Author

Vincenzo Librandi, Mar 31 2016

Keywords

References

  • Continued fraction expansion of (1651+sqrt(3236405))/2386. - Bruno Berselli, Mar 31 2016

Links

Crossrefs

Cf. A201912 (11th row of the triangle).
Cf. similar sequences of the type 2^n mod p, where p is a prime: A000034 (p=3), A070402 (p=5), A069705 (p=7), A036117 (p=11), A036118 (p=13), A062116 (p=17), A036120 (p=19), A070335 (p=23), A036122 (p=29), this sequence (p=31), A036124 (p=37), A070348 (p=41), A070349 (p=43), A070351 (p=47), A036128 (p=53), A036129 (p=59), A036130 (p=61), A036131 (p=67).

Programs

  • GAP
    List([0..70],n->PowerMod(2,n,31)); # Muniru A Asiru, Jan 30 2019
  • Magma
    [Modexp(2, n, 31): n in [0..100]];
  • Magma
    &cat [[1,2,4,8,16]^^20] // Bruno Berselli, Mar 31 2016
  • Mathematica
    PowerMod[2, Range[0, 100], 31]
  • PARI
    a(n)=2^(n%5) \\ Charles R Greathouse IV, Mar 31 2016
  • PARI
    x='x+O('x^99); Vec((1+2*x+4*x^2+8*x^3+16*x^4)/(1-x^5)) \\ Altug Alkan, Mar 31 2016
  • Python
    for n in range(0,100):print(2**n%31) # Soumil Mandal, Apr 03 2016
  • Python
    def A269266(n): return pow(2,n,31) # Chai Wah Wu, Jan 03 2022
  • Sage
    [2^mod(n,5) for n in (0..100)] # Bruno Berselli, Mar 31 2016

Formula

G.f.: (1 + 2*x + 4*x^2 + 8*x^3 + 16*x^4)/(1 - x^5).
a(n) = a(n-5).
a(n) = 2^(n mod 5). - Bruno Berselli, Mar 31 2016
Showing 1-3 of 3 results.

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