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.

Previous Showing 11-15 of 15 results.

A047441 Numbers that are congruent to {0, 2, 5, 6} mod 8.

Original entry on oeis.org

0, 2, 5, 6, 8, 10, 13, 14, 16, 18, 21, 22, 24, 26, 29, 30, 32, 34, 37, 38, 40, 42, 45, 46, 48, 50, 53, 54, 56, 58, 61, 62, 64, 66, 69, 70, 72, 74, 77, 78, 80, 82, 85, 86, 88, 90, 93, 94, 96, 98, 101, 102, 104, 106, 109, 110, 112, 114, 117, 118, 120, 122, 125
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047615, A130824.

Programs

Formula

G.f.: x^2*(2+3*x+x^2+2*x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 07 2011
From Wesley Ivan Hurt, May 26 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n-7-i^(2*n)-i^(1-n)+i^(1+n))/4 where i=sqrt(-1).
a(2k) = A130824(k) k>0, a(2k-1) = A047615(k). (End)
E.g.f.: (4 - sin(x) + (4*x - 3)*sinh(x) + 4*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 27 2016
a(n) = (8*n-7-cos(n*Pi)-2*sin(n*Pi/2))/4. - Wesley Ivan Hurt, Oct 05 2017
Sum_{n>=2} (-1)^n/a(n) = (sqrt(2)-1)*Pi/16 + (4-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8. - Amiram Eldar, Dec 21 2021

A047479 Numbers that are congruent to {0, 1, 5, 7} mod 8.

Original entry on oeis.org

0, 1, 5, 7, 8, 9, 13, 15, 16, 17, 21, 23, 24, 25, 29, 31, 32, 33, 37, 39, 40, 41, 45, 47, 48, 49, 53, 55, 56, 57, 61, 63, 64, 65, 69, 71, 72, 73, 77, 79, 80, 81, 85, 87, 88, 89, 93, 95, 96, 97, 101, 103, 104, 105, 109, 111, 112, 113, 117, 119, 120, 121, 125
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047522, A047615.

Programs

  • Magma
    I:=[0, 1, 5, 7, 8]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, May 16 2012
  • Maple
    A047479:=n->(-7-I^(2*n)+(2-I)*(-I)^n+(2+I)*I^n+8*n)/4: seq(A047479(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016
  • Mathematica
    Select[Range[0,300], MemberQ[{0,1,5,7}, Mod[#,8]]&] (* Vincenzo Librandi, May 16 2012 *)
  • PARI
    my(x='x+O('x^100)); concat(0, Vec(x^2*(1+4*x+2*x^2+x^3)/((1-x)^2*(1+x)*(1+x^2)))) \\ Altug Alkan, Dec 24 2015

Formula

From Colin Barker, May 14 2012: (Start)
a(n) = (-7-(-1)^n+(2-i)*(-i)^n+(2+i)*i^n+8*n)/4 where i=sqrt(-1).
G.f.: x^2*(1+4*x+2*x^2+x^3)/((1-x)^2*(1+x)*(1+x^2)). (End)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5. - Vincenzo Librandi, May 16 2012
a(2k) = A047522(k), a(2k-1) = A047615(k). - Wesley Ivan Hurt, Jun 01 2016
E.g.f.: (2 - sin(x) + 2*cos(x) + (4*x - 3)*sinh(x) + 4*(x - 1)*cosh(x))/2. - Ilya Gutkovskiy, Jun 02 2016
Sum_{n>=2} (-1)^n/a(n) = (sqrt(2)-1)*Pi/16 + (8-3*sqrt(2))*log(2)/16 + 3*sqrt(2)*log(2+sqrt(2))/8. - Amiram Eldar, Dec 20 2021

A047485 Numbers that are congruent to {0, 3, 5, 7} mod 8.

Original entry on oeis.org

0, 3, 5, 7, 8, 11, 13, 15, 16, 19, 21, 23, 24, 27, 29, 31, 32, 35, 37, 39, 40, 43, 45, 47, 48, 51, 53, 55, 56, 59, 61, 63, 64, 67, 69, 71, 72, 75, 77, 79, 80, 83, 85, 87, 88, 91, 93, 95, 96, 99, 101, 103, 104, 107, 109, 111, 112, 115, 117, 119, 120, 123, 125
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A004767, A047615.

Programs

Formula

From Colin Barker, May 14 2012: (Start)
G.f.: x^2*(3+2*x+2*x^2+x^3)/((1-x)^2*(1+x)*(1+x^2)).
a(n) = (-5+(-1)^n-i*(-i)^n+i*i^n+8*n)/4 where i=sqrt(-1). (End)
From Wesley Ivan Hurt, Jun 04 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(2k) = A004767(k-1) for n>0, a(2k-1) = A047615(k). (End)
E.g.f.: (2 - sin(x) + (4*x - 3)*sinh(x) + (4*x - 2)*cosh(x))/2. - Ilya Gutkovskiy, Jun 04 2016
Sum_{n>=2} (-1)^n/a(n) = (8-sqrt(2))*log(2)/16 + sqrt(2)*log(2+sqrt(2))/8 - (3-sqrt(2))*Pi/16. - Amiram Eldar, Dec 23 2021

A047492 Numbers that are congruent to {0, 4, 5, 7} mod 8.

Original entry on oeis.org

0, 4, 5, 7, 8, 12, 13, 15, 16, 20, 21, 23, 24, 28, 29, 31, 32, 36, 37, 39, 40, 44, 45, 47, 48, 52, 53, 55, 56, 60, 61, 63, 64, 68, 69, 71, 72, 76, 77, 79, 80, 84, 85, 87, 88, 92, 93, 95, 96, 100, 101, 103, 104, 108, 109, 111, 112, 116, 117, 119, 120, 124
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047535, A047615.

Programs

  • Magma
    [n : n in [0..150] | n mod 8 in [0, 4, 5, 7]]; // Wesley Ivan Hurt, May 26 2016
  • Maple
    A047492:=n->2*n+(1+I)*(2*I-2+(1-I)*I^(2*n)-I^(-n)+I^(1+n))/4: seq(A047492(n), n=1..100); # Wesley Ivan Hurt, May 26 2016
  • Mathematica
    Table[2n+(1+I)*(2*I-2+(1-I)*I^(2n)-I^(-n)+I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, May 26 2016 *)
a[n_] := 1 + n + Floor[n/2] + 2 Floor[(n - 2)/4];
Table[a[n], {n, 1, 62}] (* Peter Luschny, Dec 23 2021 *)

Formula

G.f.: x^2*(4+x+2*x^2+x^3) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Nov 06 2015
From Wesley Ivan Hurt, May 26 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = 2*n+(1+i)*(2*i-2+(1-i)*i^(2*n)-i^(-n)+i^(1+n))/4 where i=sqrt(-1).
a(2k) = A047535(k), a(2k-1) = A047615(k). (End)
E.g.f.: (2 - sin(x) - cos(x) + (4*x - 3)*sinh(x) + (4*x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 27 2016
Sum_{n>=2} (-1)^n/a(n) = (2-sqrt(2))*log(2)/8 + sqrt(2)*log(2+sqrt(2))/4 - Pi/8. - Amiram Eldar, Dec 23 2021

A382865 Bitwise XOR of all integers between n and 2n (endpoints included).

Original entry on oeis.org

0, 3, 5, 4, 8, 15, 13, 8, 16, 27, 21, 28, 24, 23, 29, 16, 32, 51, 37, 52, 40, 63, 45, 56, 48, 43, 53, 44, 56, 39, 61, 32, 64, 99, 69, 100, 72, 111, 77, 104, 80, 123, 85, 124, 88, 119, 93, 112, 96, 83, 101, 84, 104, 95, 109, 88, 112, 75, 117, 76, 120, 71, 125, 64, 128, 195
Offset: 0

Views

Author

Federico Provvedi, May 21 2025

Keywords

Examples

			a(3) = 3 XOR 4 XOR 5 XOR 6 = 4, in binary representation is: ((011 XOR 100) XOR 101) XOR 110 = (111 XOR 101) XOR 110 = 010 XOR 110 = 100 (4 in decimal).

Links

Crossrefs

Cf. A038712, A047615, A065621, A114389, A010873, A048724, A174091, A181983.

Programs

  • Maple
    a:= proc(n) option remember; uses Bits; `if`(n=0, 0,
      Xor(Xor(Xor(a(n-1), n-1), 2*n-1), 2*n))
    end:
seq(a(n), n=0..65);  # Alois P. Heinz, May 26 2025
  • Mathematica
    a[n_] = BitXor[BitOr[n-1, 2] - (-1)^n*(n-1), 4*n]/2; Table[a[n],{n,0, 65}]
  • PARI
    a(n) = my(b=n); for (i=n+1, 2*n, b = bitxor(b, i)); b; \\ Michel Marcus, May 25 2025
  • Python
    def A382865(n): return [0, n, 1, n-1][n%4] ^ (2*n) # Karl-Heinz Hofmann, May 26 2025

Formula

a(2n) = A047615(n+1), and for every integer k>1: a(n*2^k -1) = 2^k * A065621(n).
a(4n) = 8*n, a(4n+1) = 2*A114389(n+1) + 1, a(4n+2) = 8*n + 5, a(4n+3) = 4*A065621(n+1).
From Karl-Heinz Hofmann, May 27 2025: (Start)
For all n == 0 (mod 4) --> a(n) = A005843(n) = 2*n
For all n == 1 (mod 4) --> a(n) = A048724(n)
For all n == 2 (mod 4) --> a(n) = A005408(n) = 2*n + 1
For all n == 3 (mod 4) --> a(n) = A048724(n) - 1 (End)
a(n) = (1/2) * XOR(A174091(n-1) - A181983(n-1), 4*n). - Federico Provvedi, May 31 2025
Previous Showing 11-15 of 15 results.

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

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