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 1-6 of 6 results.

A047328 Numbers that are congruent to {0, 3, 5, 6} mod 7.

Original entry on oeis.org

0, 3, 5, 6, 7, 10, 12, 13, 14, 17, 19, 20, 21, 24, 26, 27, 28, 31, 33, 34, 35, 38, 40, 41, 42, 45, 47, 48, 49, 52, 54, 55, 56, 59, 61, 62, 63, 66, 68, 69, 70, 73, 75, 76, 77, 80, 82, 83, 84, 87, 89, 90, 91, 94, 96, 97, 98, 101, 103, 104, 105, 108, 110, 111
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Indices of the odd numbers in the Padovan sequence (A000931). - Francesco Daddi, Jul 31 2011

Links

Crossrefs

Cf. A000931, A047280, A047382, A134719.

Programs

Formula

G.f.: x^2*(3+2x+x^2+x^3)/((1-x)^2*(1+x)*(1+x^2)). a(n) = A028762(n-2), 2R. J. Mathar, Oct 18 2008
a(n) = (1/8)*(14*n-5-(2-(-1)^n)*(1+2*(-1)^floor(n/2))). - Bruno Berselli, Aug 01 2011
From Wesley Ivan Hurt, May 31 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14*n-7+i^(2*n)-(1+3*i)*i^(-n)-(1-3*i)*i^n)/8 where i=sqrt(-1).
a(2k) = A047280(k), a(2k-1) = A047382(k). (End)
E.g.f.: (4 - 3*sin(x) - cos(x) + (7*x - 4)*sinh(x) + (7*x - 3)*cosh(x))/4. - Ilya Gutkovskiy, May 31 2016

A047298 Numbers that are congruent to {1, 3, 4, 6} mod 7.

Original entry on oeis.org

1, 3, 4, 6, 8, 10, 11, 13, 15, 17, 18, 20, 22, 24, 25, 27, 29, 31, 32, 34, 36, 38, 39, 41, 43, 45, 46, 48, 50, 52, 53, 55, 57, 59, 60, 62, 64, 66, 67, 69, 71, 73, 74, 76, 78, 80, 81, 83, 85, 87, 88, 90, 92, 94, 95, 97, 99, 101, 102, 104, 106, 108, 109, 111
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047280, A047346.

Programs

  • Magma
    [n : n in [0..150] | n mod 7 in [1, 3, 4, 6]]; // Wesley Ivan Hurt, May 22 2016
  • Maple
    A047298:=n->I^(-n)*(I-1-7*I^n+14*n*I^n-(1+I)*I^(2*n)+I^(-n))/8: seq(A047298(n), n=1..100); # Wesley Ivan Hurt, May 22 2016
  • Mathematica
    Table[I^(-n)*(I-1-7I^n+14n*I^n-(1+I)*I^(2n)+I^(-n))/8, {n, 80}] (* Wesley Ivan Hurt, May 22 2016 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 3, 4, 6, 8}, 80] (* Vincenzo Librandi, May 24 2016 *)

Formula

a(n) = ceiling(ceiling((7n + 2)/2)/2).
G.f.: x*(1+2*x+x^2+2*x^3+x^4) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Oct 25 2011
From Wesley Ivan Hurt, May 22 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = i^(-n)*(i-1-7*i^n+14*n*i^n-(1+i)*i^(2n)+i^(-n))/8 where i=sqrt(-1).
a(2n) = A047280(n), a(2n-1) = A047346(n). (End)
E.g.f.: (4 + sin(x) - cos(x) + (7*x - 4)*sinh(x) + (7*x - 3)*cosh(x))/4. - Ilya Gutkovskiy, May 23 2016

Extensions

More terms from Wesley Ivan Hurt, May 22 2016

A047329 Numbers that are congruent to {1, 3, 5, 6} mod 7.

Original entry on oeis.org

1, 3, 5, 6, 8, 10, 12, 13, 15, 17, 19, 20, 22, 24, 26, 27, 29, 31, 33, 34, 36, 38, 40, 41, 43, 45, 47, 48, 50, 52, 54, 55, 57, 59, 61, 62, 64, 66, 68, 69, 71, 73, 75, 76, 78, 80, 82, 83, 85, 87, 89, 90, 92, 94, 96, 97, 99, 101, 103, 104, 106, 108, 110, 111
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • Robert Fludd, Utriusque Cosmi ... Historia, Oppenheim, 1617-1619.

Links

Crossrefs

Cf. A047280, A047383, A082977.

Programs

Formula

a(n) = floor((7n-1)/4). - Gary Detlefs, Mar 07 2010
G.f.: (x*(1+2*x+2*x^2+x^3+x^4)) / ((1+x)*(x^2+1)*(x-1)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = a(n-1)+a(n-4)-a(n-5) for n>5.
a(n) = (14n-5-i^(2n)-(1+i)*i^(-n)-(1-i)*i^n)/8 where i=sqrt(-1).
a(2n) = A047280(n), a(2n-1) = A047383(n). (End)
E.g.f.: (4 - sin(x) - cos(x) + (7*x - 2)*sinh(x) + (7*x - 3)*cosh(x))/4. - Ilya Gutkovskiy, May 21 2016

Extensions

Fludd reference from Brendan McKay, May 27 2003
More terms from Wesley Ivan Hurt, May 21 2016

A047297 Numbers that are congruent to {0, 3, 4, 6} mod 7.

Original entry on oeis.org

0, 3, 4, 6, 7, 10, 11, 13, 14, 17, 18, 20, 21, 24, 25, 27, 28, 31, 32, 34, 35, 38, 39, 41, 42, 45, 46, 48, 49, 52, 53, 55, 56, 59, 60, 62, 63, 66, 67, 69, 70, 73, 74, 76, 77, 80, 81, 83, 84, 87, 88, 90, 91, 94, 95, 97, 98, 101, 102, 104, 105, 108, 109, 111
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047280, A047345.

Programs

  • Magma
    [n : n in [0..150] | n mod 7 in [0, 3, 4, 6]]; // Wesley Ivan Hurt, Jun 02 2016
  • Maple
    A047297:=n->(14*n-9+3*I^(2*n)-(1+I)*I^(-n)-(1-I)*I^n)/8: seq(A047297(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016
  • Mathematica
    Table[(14n-9+3*I^(2n)-(1+I)*I^(-n)-(1-I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, Jun 02 2016 *)

Formula

G.f.: x^2*(3+x+2*x^2+x^3) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 25 2011
From Wesley Ivan Hurt, Jun 02 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14*n-9+3*i^(2*n)-(1+i)*i^(-n)-(1-i)*i^n)/8 where i=sqrt(-1).
a(2k) = A047280(k), a(2k-1) = A047345(k). (End)

A047300 Numbers that are congruent to {2, 3, 4, 6} mod 7.

Original entry on oeis.org

2, 3, 4, 6, 9, 10, 11, 13, 16, 17, 18, 20, 23, 24, 25, 27, 30, 31, 32, 34, 37, 38, 39, 41, 44, 45, 46, 48, 51, 52, 53, 55, 58, 59, 60, 62, 65, 66, 67, 69, 72, 73, 74, 76, 79, 80, 81, 83, 86, 87, 88, 90, 93, 94, 95, 97, 100, 101, 102, 104, 107, 108, 109, 111
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047280, A047348.

Programs

  • Magma
    [n : n in [0..150] | n mod 7 in [2, 3, 4, 6]]; // Wesley Ivan Hurt, Jun 02 2016
  • Maple
    A047300:=n->(14*n-5-I^(2*n)-(1-3*I)*I^(-n)-(1+3*I)*I^n)/8: seq(A047300(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016
  • Mathematica
    Table[(14n-5-I^(2n)-(1-3*I)*I^(-n)-(1+3*I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, Jun 02 2016 *)

Formula

G.f.: x*(2+x+x^2+2*x^3+x^4) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Oct 25 2011
From Wesley Ivan Hurt, Jun 02 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14*n-5-i^(2*n)-(1-3*i)*i^(-n)-(1+3*i)*i^n)/8 where i=sqrt(-1).
a(2k) = A047280(k), a(2k-1) = A047348(k). (End)

A047331 Numbers that are congruent to {2, 3, 5, 6} mod 7.

Original entry on oeis.org

2, 3, 5, 6, 9, 10, 12, 13, 16, 17, 19, 20, 23, 24, 26, 27, 30, 31, 33, 34, 37, 38, 40, 41, 44, 45, 47, 48, 51, 52, 54, 55, 58, 59, 61, 62, 65, 66, 68, 69, 72, 73, 75, 76, 79, 80, 82, 83, 86, 87, 89, 90, 93, 94, 96, 97, 100, 101, 103, 104, 107, 108, 110, 111
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047280, A047385.

Programs

  • Magma
    [n : n in [0..150] | n mod 7 in [2, 3, 5, 6]]; // Wesley Ivan Hurt, Jun 03 2016
  • Maple
    A047331:=n->(14*n-3-3*I^(2*n)-(1-I)*I^(-n)-(1+I)*I^n)/8: seq(A047331(n), n=1..100); # Wesley Ivan Hurt, Jun 03 2016
  • Mathematica
    Table[(14n-3-3*I^(2n)-(1-I)*I^(-n)-(1+I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, Jun 03 2016 *)

Formula

G.f.: x*(2+x+2*x^2+x^3+x^4) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Dec 04 2011
From Wesley Ivan Hurt, Jun 03 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14*n-3-3*i^(2*n)-(1-i)*i^(-n)-(1+i)*i^n)/8 where i=sqrt(-1).
a(2k) = A047280(k), a(2k-1) = A047385(k). (End)
Showing 1-6 of 6 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.