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.

A047285 Numbers that are congruent to {0, 2, 3, 6} mod 7.

Original entry on oeis.org

0, 2, 3, 6, 7, 9, 10, 13, 14, 16, 17, 20, 21, 23, 24, 27, 28, 30, 31, 34, 35, 37, 38, 41, 42, 44, 45, 48, 49, 51, 52, 55, 56, 58, 59, 62, 63, 65, 66, 69, 70, 72, 73, 76, 77, 79, 80, 83, 84, 86, 87, 90, 91, 93, 94, 97, 98, 100, 101, 104, 105, 107, 108, 111
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047276, A047355.

Programs

  • Magma
    [n : n in [0..150] | n mod 7 in [0, 2, 3, 6]]; // Wesley Ivan Hurt, Jun 02 2016
  • Maple
    A047285:=n->(14*n-13+3*I^(2*n)+(1+I)*I^(-n)+(1-I)*I^n)/8: seq(A047285(n), n=1..100); # Wesley Ivan Hurt, Jun 02 2016
  • Mathematica
    Table[(14n-13+3*I^(2n)+(1+I)*I^(-n)+(1-I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, Jun 02 2016 *)
Select[Range[0,120],MemberQ[{0,2,3,6},Mod[#,7]]&] (* or *) LinearRecurrence[ {1,0,0,1,-1},{0,2,3,6,7},100] (* Harvey P. Dale, Jul 12 2020 *)

Formula

G.f.: x^2*(2+x+3*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-13+3*i^(2*n)+(1+i)*i^(-n)+(1-i)*i^n)/8 where i=sqrt(-1).
a(2k) = A047276(k), a(2k-1) = A047355(k). (End)

A047286 Numbers that are congruent to {1, 2, 3, 6} mod 7.

Original entry on oeis.org

1, 2, 3, 6, 8, 9, 10, 13, 15, 16, 17, 20, 22, 23, 24, 27, 29, 30, 31, 34, 36, 37, 38, 41, 43, 44, 45, 48, 50, 51, 52, 55, 57, 58, 59, 62, 64, 65, 66, 69, 71, 72, 73, 76, 78, 79, 80, 83, 85, 86, 87, 90, 92, 93, 94, 97, 99, 100, 101, 104, 106, 107, 108, 111
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047276, A047356.

Programs

  • Magma
    [n : n in [0..150] | n mod 7 in [1, 2, 3, 6]]; // Wesley Ivan Hurt, May 22 2016
  • Maple
    A047286:=n->(14*n-11+I^(2*n)+(1+3*I)*I^(-n)+(1-3*I)*I^n)/8: seq(A047286(n), n=1..100); # Wesley Ivan Hurt, May 22 2016
  • Mathematica
    Table[(14n-11+I^(2n)+(1+3I)*I^(-n)+(1-3I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, May 22 2016 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 2, 3, 6, 8}, 80] (* Vincenzo Librandi, May 24 2016 *)

Formula

G.f.: x*(1+x+x^2+3*x^3+x^4) / ( (1+x)*(1+x^2)*(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) = (14n-11+i^(2n)+(1+3i)*i^(-n)+(1-3i)*i^n)/8 where i=sqrt(-1).
a(2n) = A047276(n), a(2n-1) = A047356(n). (End)
E.g.f.: (4 + 3*sin(x) + cos(x) + (7*x - 6)*sinh(x) + (7*x - 5)*cosh(x))/4. - Ilya Gutkovskiy, May 23 2016

Extensions

More terms from Wesley Ivan Hurt, May 22 2016

A047293 Numbers that are congruent to {0, 2, 4, 6} mod 7.

Original entry on oeis.org

0, 2, 4, 6, 7, 9, 11, 13, 14, 16, 18, 20, 21, 23, 25, 27, 28, 30, 32, 34, 35, 37, 39, 41, 42, 44, 46, 48, 49, 51, 53, 55, 56, 58, 60, 62, 63, 65, 67, 69, 70, 72, 74, 76, 77, 79, 81, 83, 84, 86, 88, 90, 91, 93, 95, 97, 98, 100, 102, 104, 105, 107, 109, 111
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047276, A047345.

Programs

Formula

a(n) = floor(ceiling((7n + 2)/2)/2).
a(n) = 2n-2-floor((n-1)/4). - Gary Detlefs, Mar 27 2010
From Colin Barker, Mar 13 2012: (Start)
a(n) = a(n-1)+a(n-4)-a(n-5) for n>5.
G.f.: x^2*(2+2*x+2*x^2+x^3)/((1-x)^2*(1+x)*(1+x^2)). (End)
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = (14n-11+i^(2n)+(1-i)*i^(-n)+(1+i)*i^n)/8 where i=sqrt(-1).
a(2n) = A047276(n), a(2n-1) = A047345(n). (End)

A047294 Numbers that are congruent to {1, 2, 4, 6} mod 7.

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 11, 13, 15, 16, 18, 20, 22, 23, 25, 27, 29, 30, 32, 34, 36, 37, 39, 41, 43, 44, 46, 48, 50, 51, 53, 55, 57, 58, 60, 62, 64, 65, 67, 69, 71, 72, 74, 76, 78, 79, 81, 83, 85, 86, 88, 90, 92, 93, 95, 97, 99, 100, 102, 104, 106, 107, 109, 111
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047276, A047346.

Programs

  • Magma
    I:=[1, 2, 4, 6, 8]; [n le 5 select I[n] else Self(n-1)+Self(n-4)-Self(n-5): n in [1..70]]; // Vincenzo Librandi, Apr 27 2012
  • Maple
    A047294:=n->ceil(floor((7*n-5)/2)/2): seq(A047294(n), n=1..100); # Wesley Ivan Hurt, Jun 01 2016
  • Mathematica
    Select[Range[0,100], MemberQ[{1,2,4,6}, Mod[#,7]]&] (* Vincenzo Librandi, Apr 27 2012 *)
LinearRecurrence[{1,0,0,1,-1},{1,2,4,6,8},100] (* G. C. Greubel, Jun 01 2016 *)
  • PARI
    x='x+O('x^100); Vec(x*(1+x+2*x^2+2*x^3+x^4)/((1-x)^2*(1+x)*(1+x^2))) \\ Altug Alkan, Dec 24 2015

Formula

a(n) = ceiling(floor((7*n - 5)/2)/2).
From Colin Barker, Mar 14 2012: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
G.f.: x*(1 + x + 2*x^2 + 2*x^3 + x^4)/((1-x)^2*(1+x)*(1+x^2)). (End)
a(n) = (-9 -(-1)^n + (1+i)*(-i)^n + (1-i)*i^n + 14*n)/8 where i=sqrt(-1). - Colin Barker, May 14 2012
a(2k) = A047276(k), a(2k-1) = A047346(k). - Wesley Ivan Hurt, Jun 01 2016
E.g.f.: (4 + sin(x) + cos(x) + (7*x - 4)*sinh(x) + (7*x - 5)*cosh(x))/4. - Ilya Gutkovskiy, Jun 01 2016

A047324 Numbers that are congruent to {0, 2, 5, 6} mod 7.

Original entry on oeis.org

0, 2, 5, 6, 7, 9, 12, 13, 14, 16, 19, 20, 21, 23, 26, 27, 28, 30, 33, 34, 35, 37, 40, 41, 42, 44, 47, 48, 49, 51, 54, 55, 56, 58, 61, 62, 63, 65, 68, 69, 70, 72, 75, 76, 77, 79, 82, 83, 84, 86, 89, 90, 91, 93, 96, 97, 98, 100, 103, 104, 105, 107, 110, 111
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047276, A047382.

Programs

  • Magma
    [n : n in [0..150] | n mod 7 in [0, 2, 5, 6]]; // Wesley Ivan Hurt, Jun 03 2016
  • Maple
    A047324:=n->(14*n-9-I^(2*n)+(1-3*I)*I^(-n)+(1+3*I)*I^n)/8: seq(A047324(n), n=1..100); # Wesley Ivan Hurt, Jun 03 2016
  • Mathematica
    Table[(14n - 9 - I^(2n) + (1 - 3 * I) * I^(-n) + (1 + 3 * I) * I^n)/8, {n, 80}] (* Wesley Ivan Hurt, Jun 03 2016 *)
Flatten[Table[7n + {0, 2, 5, 6}, {n, 0, 15}]] (* Alonso del Arte, Jun 04 2016 *)
LinearRecurrence[{1,0,0,1,-1},{0,2,5,6,7},80] (* Harvey P. Dale, Jan 10 2023 *)

Formula

G.f.: x^2*(2+3*x+x^2+x^3) / ( (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 - 9 - i^(2*n) + (1 - 3*i)*i^(-n) + (1 + 3*i)*i^n)/8 where i = sqrt(-1).
a(2k) = A047276(k), a(2k-1) = A047382(k). (End)
E.g.f.: (4 - 3*sin(x) + cos(x) + (7*x - 4)*sinh(x) + (7*x - 5)*cosh(x))/4. - Ilya Gutkovskiy, Jun 04 2016

A047325 Numbers that are congruent to {1, 2, 5, 6} mod 7.

Original entry on oeis.org

1, 2, 5, 6, 8, 9, 12, 13, 15, 16, 19, 20, 22, 23, 26, 27, 29, 30, 33, 34, 36, 37, 40, 41, 43, 44, 47, 48, 50, 51, 54, 55, 57, 58, 61, 62, 64, 65, 68, 69, 71, 72, 75, 76, 78, 79, 82, 83, 85, 86, 89, 90, 92, 93, 96, 97, 99, 100, 103, 104, 106, 107, 110, 111
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047276, A047383.

Programs

  • Magma
    [n : n in [0..150] | n mod 7 in [1, 2, 5, 6]]; // Wesley Ivan Hurt, May 23 2016
  • Maple
    A047325:=n->(14*n-7-3*I^(2*n)+(1-I)*I^(-n)+(1+I)*I^n)/8: seq(A047325(n), n=1..100); # Wesley Ivan Hurt, May 23 2016
  • Mathematica
    Table[(14n-7-3*I^(2n)+(1-I)*I^(-n)+(1+I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, May 23 2016 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {1, 2, 5, 6, 8}, 80] (* Vincenzo Librandi, May 24 2016 *)
#+{1,2,5,6}&/@(7*Range[0,20])//Flatten (* Harvey P. Dale, Aug 16 2018 *)

Formula

G.f.: x*(1+x+3*x^2+x^3+x^4) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Dec 04 2011
From Wesley Ivan Hurt, May 23 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n>5.
a(n) = (14n-7-3*i^(2n)+(1-i)*i^(-n)+(1+i)*i^n)/8 where i=sqrt(-1).
a(2n) = A047276(n), a(2n-1) = A047383(n). (End)
E.g.f.: (4 - sin(x) + cos(x) + (7*x - 2)*sinh(x) + (7*x - 5)*cosh(x))/4. - Ilya Gutkovskiy, May 24 2016

Extensions

More terms from Wesley Ivan Hurt, May 23 2016
Showing 1-6 of 6 results.

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

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