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 1-4 of 4 results.

A047621 Numbers that are congruent to {3, 5} mod 8.

Original entry on oeis.org

3, 5, 11, 13, 19, 21, 27, 29, 35, 37, 43, 45, 51, 53, 59, 61, 67, 69, 75, 77, 83, 85, 91, 93, 99, 101, 107, 109, 115, 117, 123, 125, 131, 133, 139, 141, 147, 149, 155, 157, 163, 165, 171, 173, 179, 181, 187, 189, 195, 197, 203, 205, 211, 213, 219, 221, 227, 229
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Numbers k for which Jacobi symbol J(2,k) = -1, so 2 (as well as 2^k) is not a square mod k. - Antti Karttunen, Aug 27 2005, corrected by Jianing Song, Nov 05 2019, see also A329095.
Numbers n whose multiplicative order modulo 2^k is 2^(k - 2) for k >= 4. For k = 3, the numbers whose multiplicative order modulo 8 is 2 are in sequence A047484. - Jianing Song, Apr 29 2018

Links

Crossrefs

Row 1 of A112070. Complement of A047522 relative to A005408. Primes in this sequence: A003629.
Subsequence of A329095.
Cf. A047522, A066507, A101464, A144981.

Programs

  • GAP
    a:=[3];; for n in [2..60] do a[n]:=8*n-a[n-1]-8; od; a; # Muniru A Asiru, Dec 04 2018
  • Haskell
    a047621 n = a047621_list !! (n-1)
a047621_list = 3 : 5 : map (+ 8) a047621_list
-- Reinhard Zumkeller, Jul 05 2013
  • Mathematica
    LinearRecurrence[{1, 1, -1}, {3, 5, 11}, 100] (* Jean-François Alcover, Jul 31 2018 *)

Formula

a(n) = 8*n - a(n-1) - 8 (with a(1) = 3). - Vincenzo Librandi, Aug 06 2010
G.f.: x*(3 + 2*x + 3*x^2) / ( (1 + x)*(x - 1)^2 ). - R. J. Mathar, Oct 08 2011
A089911(3*a(n)) = 10. - Reinhard Zumkeller, Jul 05 2013
a(n) = 8*floor((n - 1)/2) + 4 + (-1)^n. - Gary Detlefs, Dec 03 2018
From Franck Maminirina Ramaharo, Dec 03 2018: (Start)
a(n) = 4*n - 2 - (-1)^n.
E.g.f.: 3 - (2 - 4*x)*exp(x) - exp(-x). (End)
a(n + 2) = a(n) + 8. - David A. Corneth, Dec 03 2018
Sum_{n>=1} (-1)^(n+1)/a(n) = (sqrt(2)-1)*Pi/8. - Amiram Eldar, Dec 11 2021
From Amiram Eldar, Nov 22 2024: (Start)
Product_{n>=1} (1 - (-1)^n/a(n)) = sec(Pi/8) (1/A144981).
Product_{n>=1} (1 + (-1)^n/a(n)) = 2*sin(Pi/8) (A101464). (End)

A047529 Numbers that are congruent to {1, 3, 7} mod 8.

Original entry on oeis.org

1, 3, 7, 9, 11, 15, 17, 19, 23, 25, 27, 31, 33, 35, 39, 41, 43, 47, 49, 51, 55, 57, 59, 63, 65, 67, 71, 73, 75, 79, 81, 83, 87, 89, 91, 95, 97, 99, 103, 105, 107, 111, 113, 115, 119, 121, 123, 127, 129, 131, 135, 137, 139, 143, 145, 147, 151, 153, 155, 159
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Terms that occur on the first two rows of array A257852. Odd numbers that are not of the form 4k+1, where k is an odd number. - Antti Karttunen, Jun 06 2024

Examples

			G.f. = x + 3*x^2 + 7*x^3 + 9*x^4 + 11*x^5 + 15*x^6 + 17*x^7 + 19*x^8 + 23*x^9 + ...

Links

Crossrefs

Cf. A047478, A047484, A047623.
Setwise difference A005408 \ A004770.
Disjoint union of A004767 and A017077; see A257852.

Programs

  • Magma
    [n : n in [0..150] | n mod 8 in [1, 3, 7]]; // Wesley Ivan Hurt, Jun 13 2016
  • Maple
    A047529:=n->(24*n+2*sqrt(3)*sin(2*Pi*n/3)+6*cos(2*Pi*n/3)-15)/9: seq(A047529(n), n=1..100); # Wesley Ivan Hurt, Jun 13 2016
  • Mathematica
    Select[Range[150], MemberQ[{1,3,7}, Mod[#,8]]&] (* Harvey P. Dale, May 02 2011 *)
LinearRecurrence[{1, 0, 1, -1}, {1, 3, 7, 9}, 100] (* Vincenzo Librandi, Jun 14 2016 *)
  • PARI
    Vec(x*(x^3+4*x^2+2*x+1)/((x-1)^2*(x^2+x+1)) + O(x^100)) \\ Colin Barker, Nov 12 2015
  • PARI
    {a(n) = n\3 * 8 + [-1, 1, 3][n%3 + 1]}; /* Michael Somos, Nov 15 2015 */

Formula

a(n) = (24*n+2*sqrt(3)*sin(2*Pi*n/3)+6*cos(2*Pi*n/3)-15)/9. - Fred Daniel Kline, Nov 12 2015
From Colin Barker, Nov 12 2015: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4.
G.f.: x*(x^3+4*x^2+2*x+1) / ((x-1)^2*(x^2+x+1)). (End)
a(n+3) = a(n) + 8 for all n in Z. - Michael Somos, Nov 15 2015
a(3k) = 8k-1, a(3k-1) = 8k-5, a(3k-2) = 8k-7. - Wesley Ivan Hurt, Jun 13 2016
a(n) = 8 * floor((n-1) / 3) + 2^(((n-1) mod 3) + 1) - 1. - Fred Daniel Kline, Aug 09 2016
a(n) = 2*(n + floor(n/3)) - 1. - Wolfdieter Lang, Sep 10 2021

A047478 Numbers that are congruent to {1, 5, 7} mod 8.

Original entry on oeis.org

1, 5, 7, 9, 13, 15, 17, 21, 23, 25, 29, 31, 33, 37, 39, 41, 45, 47, 49, 53, 55, 57, 61, 63, 65, 69, 71, 73, 77, 79, 81, 85, 87, 89, 93, 95, 97, 101, 103, 105, 109, 111, 113, 117, 119, 121, 125, 127, 129, 133, 135, 137, 141, 143, 145, 149, 151, 153, 157, 159
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047484, A047529, A047623.

Programs

  • Magma
    I:=[1, 5, 7, 9]; [n le 4 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..70]]; // Vincenzo Librandi, May 16 2012
  • Maple
    A047478:=n->(24*n-9-4*sqrt(3)*sin(2*n*Pi/3))/9: seq(A047478(n), n=1..100); # Wesley Ivan Hurt, Jun 10 2016
  • Mathematica
    Select[Range[0,300], MemberQ[{1,5,7}, Mod[#,8]]&] (* Vincenzo Librandi, May 16 2012 *)

Formula

G.f.: x*(1+4*x+2*x^2+x^3)/((1-x)^2*(1+x+x^2)). [Colin Barker, May 14 2012]
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4. - Vincenzo Librandi, May 16 2012
From Wesley Ivan Hurt, Jun 10 2016: (Start)
a(n) = (24*n-9-4*sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 8k-1, a(3k-1) = 8k-3, a(3k-2) = 8k-7. (End)
a(n) = 2*(n + floor((n+1)/3)) - 1. - Wolfdieter Lang, Sep 11 2021

A047623 Numbers that are congruent to {1, 3, 5} mod 8.

Original entry on oeis.org

1, 3, 5, 9, 11, 13, 17, 19, 21, 25, 27, 29, 33, 35, 37, 41, 43, 45, 49, 51, 53, 57, 59, 61, 65, 67, 69, 73, 75, 77, 81, 83, 85, 89, 91, 93, 97, 99, 101, 105, 107, 109, 113, 115, 117, 121, 123, 125, 129, 131, 133, 137, 139, 141, 145, 147, 149, 153, 155, 157
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Numbers that can be expressed as the sum of at most three square numbers (see Tattersall). - Stefano Spezia, Jul 02 2025

References

  • James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 17.

Links

Crossrefs

Cf. A047478, A047484, A047529.

Programs

  • Magma
    I:=[1, 3, 5, 9]; [n le 4 select I[n] else Self(n-1)+Self(n-3)-Self(n-4): n in [1..70]]; // Vincenzo Librandi, Apr 27 2012
  • Maple
    A047623:=n->(24*n-21-6*cos(2*n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/9: seq(A047623(n), n=1..100); # Wesley Ivan Hurt, Jun 10 2016
  • Mathematica
    Select[Range[0,150], MemberQ[{1,3,5}, Mod[#,8]]&] (* Vincenzo Librandi, Apr 27 2012 *)

Formula

a(n) = 2*floor((n-1)/3) + 2*n - 1. - Gary Detlefs, Mar 18 2010
From Colin Barker, Feb 03 2012: (Start)
G.f.: x*(1+2*x+2*x^2+3*x^3)/(1-x-x^3+x^4).
a(n) = a(n-1) + a(n-3) - a(n-4) for n>4. (End)
From Wesley Ivan Hurt, Jun 10 2016: (Start)
a(n) = (24*n-21-6*cos(2*n*Pi/3)+2*sqrt(3)*sin(2*n*Pi/3))/9.
a(3k) = 8k-3, a(3k-1) = 8k-5, a(3k-2) = 8k-7. (End)
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.