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.

A047417 Numbers that are congruent to {2, 3, 4, 6} mod 8.

Original entry on oeis.org

2, 3, 4, 6, 10, 11, 12, 14, 18, 19, 20, 22, 26, 27, 28, 30, 34, 35, 36, 38, 42, 43, 44, 46, 50, 51, 52, 54, 58, 59, 60, 62, 66, 67, 68, 70, 74, 75, 76, 78, 82, 83, 84, 86, 90, 91, 92, 94, 98, 99, 100, 102, 106, 107, 108, 110, 114, 115, 116, 118, 122, 123
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047398, A047463.

Programs

Formula

G.f.: x*(2+x+x^2+2*x^3+2*x^4) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 05 2011
From Wesley Ivan Hurt, May 25 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n-5-i^(2*n)-(1-2*i)*i^(-n)-(1+2*i)*i^n)/4 with i=sqrt(-1).
a(2k) = A047398(k), a(2k-1) = A047463(k). (End)
E.g.f.: 2 + sin(x) - cos(x)/2 + (2*x - 1)*sinh(x) + (2*x - 3/2)*cosh(x). - Ilya Gutkovskiy, May 25 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (3-sqrt(2))*Pi/16 - (sqrt(2)+4)*log(2)/16 + sqrt(2)*log(sqrt(2)+2)/8. - Amiram Eldar, Dec 25 2021

A047444 Numbers that are congruent to {0, 3, 5, 6} mod 8.

Original entry on oeis.org

0, 3, 5, 6, 8, 11, 13, 14, 16, 19, 21, 22, 24, 27, 29, 30, 32, 35, 37, 38, 40, 43, 45, 46, 48, 51, 53, 54, 56, 59, 61, 62, 64, 67, 69, 70, 72, 75, 77, 78, 80, 83, 85, 86, 88, 91, 93, 94, 96, 99, 101, 102, 104, 107, 109, 110, 112, 115, 117, 118, 120, 123, 125
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047398, A047645.

Programs

  • Magma
    [n : n in [0..150] | n mod 8 in [0, 3, 5, 6]]; // Wesley Ivan Hurt, May 26 2016
  • Maple
    A047444:=n->(1+I)*(4*n-4*n*I+3*I-3-I^(-n)+I^(1+n))/4: seq(A047444(n), n=1..100); # Wesley Ivan Hurt, May 26 2016
  • Mathematica
    Table[(1+I)*(4n-4*n*I+3*I-3-I^(-n)+I^(1+n))/4, {n, 80}] (* Wesley Ivan Hurt, May 26 2016 *)
LinearRecurrence[{2,-2,2,-1},{0,3,5,6},70] (* Harvey P. Dale, Aug 26 2019 *)

Formula

G.f.: x^2*(3-x+2*x^2) / ( (x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 07 2011
From Wesley Ivan Hurt, May 26 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(n) = (1+i)*(4*n-4*n*i+3*i-3-i^(-n)+i^(1+n))/4 where i=sqrt(-1).
a(2k) = A047398(k), a(2k-1) = A047645(k). (End)
E.g.f.: (4 - sin(x) - cos(x) + (4*x - 3)*exp(x))/2. - Ilya Gutkovskiy, May 27 2016
Sum_{n>=2} (-1)^n/a(n) = 3*log(2)/8 - (3-2*sqrt(2))*Pi/16. - Amiram Eldar, Dec 21 2021

A047445 Numbers that are congruent to {1, 3, 5, 6} mod 8.

Original entry on oeis.org

1, 3, 5, 6, 9, 11, 13, 14, 17, 19, 21, 22, 25, 27, 29, 30, 33, 35, 37, 38, 41, 43, 45, 46, 49, 51, 53, 54, 57, 59, 61, 62, 65, 67, 69, 70, 73, 75, 77, 78, 81, 83, 85, 86, 89, 91, 93, 94, 97, 99, 101, 102, 105, 107, 109, 110, 113, 115, 117, 118, 121, 123, 125
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A016813, A047398.

Programs

Formula

G.f.: x*(1+2*x+2*x^2+x^3+2*x^4) / ( (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-5-i^(2*n)-i^(-n)-i^n)/4 where i=sqrt(-1).
a(2k) = A047398(k), a(2k-1) = A016813(k-1) for k>0. (End)
E.g.f.: (4 - cos(x) + (4*x - 2)*sinh(x) + (4*x - 3)*cosh(x))/2. - Ilya Gutkovskiy, May 27 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (4-sqrt(2))*Pi/16 + log(2)/8 + sqrt(2)*log(sqrt(2)+1)/8. - Amiram Eldar, Dec 24 2021

A047514 Numbers that are congruent to {3, 4, 6, 7} mod 8.

Original entry on oeis.org

3, 4, 6, 7, 11, 12, 14, 15, 19, 20, 22, 23, 27, 28, 30, 31, 35, 36, 38, 39, 43, 44, 46, 47, 51, 52, 54, 55, 59, 60, 62, 63, 67, 68, 70, 71, 75, 76, 78, 79, 83, 84, 86, 87, 91, 92, 94, 95, 99, 100, 102, 103, 107, 108, 110, 111, 115, 116, 118, 119, 123, 124
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047398, A047535.

Programs

  • Magma
    [n : n in [0..150] | n mod 8 in [3, 4, 6, 7]]; // Wesley Ivan Hurt, May 27 2016
  • Maple
    A047514:=n->(1+I)*(4*n-4*n*I+(I-1)*I^(2*n)+I^(1-n)-I^n)/4: seq(A047514(n), n=1..100); # Wesley Ivan Hurt, May 27 2016
  • Mathematica
    Table[(1+I)*(4n-4n*I+(I-1)*I^(2n)+I^(1-n)-I^n)/4, {n, 80}] (* Wesley Ivan Hurt, May 27 2016 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {3, 4, 6, 7, 11}, 100] (* Vincenzo Librandi, Aug 11 2016 *)

Formula

From Wesley Ivan Hurt, May 27 2016: (Start)
G.f.: x*(3+x+2*x^2+x^3+x^4) / ((x-1)^2*(1+x+x^2+x^3)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (1+i)*(4*n-4*n*i+(i-1)*i^(2*n)+i^(1-n)-i^n)/4 where i=sqrt(-1).
a(2k) = A047535(k), a(2k-1) = A047398(k). (End)
E.g.f.: (2 + sin(x) - cos(x) + (4*x + 1)*sinh(x) + (4*x - 1)*cosh(x))/2. - Ilya Gutkovskiy, May 27 2016
From Wesley Ivan Hurt, Aug 10 2016: (Start)
a(n) = a(n-4) + 8 for n > 4.
a(4*k) = 8*k-1, a(4*k-1) = 8*k-2, a(4*k-2) = 8*k-4, a(4*k-3) = 8*k-5. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = (2*sqrt(2)-1)*Pi/16 - (sqrt(2)-1)*log(2)/8 + sqrt(2)*log(2-sqrt(2))/4. - Amiram Eldar, Dec 26 2021

A047582 Numbers that are congruent to {3, 5, 6, 7} mod 8.

Original entry on oeis.org

3, 5, 6, 7, 11, 13, 14, 15, 19, 21, 22, 23, 27, 29, 30, 31, 35, 37, 38, 39, 43, 45, 46, 47, 51, 53, 54, 55, 59, 61, 62, 63, 67, 69, 70, 71, 75, 77, 78, 79, 83, 85, 86, 87, 91, 93, 94, 95, 99, 101, 102, 103, 107, 109, 110, 111, 115, 117, 118, 119, 123, 125
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047398, A047550.

Programs

Formula

From Wesley Ivan Hurt, May 29 2016: (Start)
G.f.: x*(3+2*x+x^2+x^3+x^4) / ((x-1)^2*(1+x+x^2+x^3)).
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (8*n+1-i^(2*n)-(2-i)*i^(-n)-(2+i)*i^n)/4 where i=sqrt(-1).
a(2k) = A047550(k), a(2k-1) = A047398(k). (End)
E.g.f.: (2 + sin(x) - 2*cos(x) + sinh(x) + 4*x*exp(x))/2. - Ilya Gutkovskiy, May 30 2016
Sum_{n>=1} (-1)^(n+1)/a(n) = (3*sqrt(2)-2)*Pi/16 - log(2)/8 + sqrt(2)*log(3-2*sqrt(2))/16. - Amiram Eldar, Dec 26 2021
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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