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.

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

A047322 Numbers that are congruent to {0, 1, 5, 6} mod 7.

Original entry on oeis.org

0, 1, 5, 6, 7, 8, 12, 13, 14, 15, 19, 20, 21, 22, 26, 27, 28, 29, 33, 34, 35, 36, 40, 41, 42, 43, 47, 48, 49, 50, 54, 55, 56, 57, 61, 62, 63, 64, 68, 69, 70, 71, 75, 76, 77, 78, 82, 83, 84, 85, 89, 90, 91, 92, 96, 97, 98, 99, 103, 104, 105, 106, 110, 111
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A030308, A047336, A047382.

Programs

  • Magma
    [n : n in [0..150] | n mod 7 in [0, 1, 5, 6]]; // Wesley Ivan Hurt, May 23 2016
  • Maple
    A047322:=n->(14*n-11-3*I^(2*n)+(3-3*I)*I^(-n)+(3+3*I)*I^n)/8: seq(A047322(n), n=1..100); # Wesley Ivan Hurt, May 23 2016
  • Mathematica
    Table[(14n-11-3*I^(2n)+(3-3*I)*I^(-n)+(3+3*I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, May 23 2016 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 1, 5, 6, 7}, 60] (* Vincenzo Librandi, May 24 2016 *)

Formula

a(n+1) = Sum_{k>=0} A030308(n,k)*b(k) with b(0)=1, b(1)=5, b(k)=7*2^(k-2) for k>1. - Philippe Deléham, Oct 19 2011
G.f.: x^2*(1+4*x+x^2+x^3) / ( (1+x)*(x^2+1)*(x-1)^2 ). - R. J. Mathar, Dec 03 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-11-3*I^(2n)+(3-3*I)*I^(-n)+(3+3*I)*I^n)/8 where I=sqrt(-1).
a(2n) = A047336(n), a(2n-1) = A047382(n). (End)
E.g.f.: (4 - 3*sin(x) + 3*cos(x) + (7*x - 4)*sinh(x) + 7*(x - 1)*cosh(x))/4. - Ilya Gutkovskiy, May 24 2016

Extensions

More terms from Wesley Ivan Hurt, May 23 2016

A047312 Numbers that are congruent to {0, 4, 5, 6} mod 7.

Original entry on oeis.org

0, 4, 5, 6, 7, 11, 12, 13, 14, 18, 19, 20, 21, 25, 26, 27, 28, 32, 33, 34, 35, 39, 40, 41, 42, 46, 47, 48, 49, 53, 54, 55, 56, 60, 61, 62, 63, 67, 68, 69, 70, 74, 75, 76, 77, 81, 82, 83, 84, 88, 89, 90, 91, 95, 96, 97, 98, 102, 103, 104, 105, 109, 110, 111
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Links

Crossrefs

Cf. A047288, A047382.

Programs

  • Magma
    [n : n in [0..150] | n mod 7 in [0, 4, 5, 6]]; // Wesley Ivan Hurt, Jun 03 2016
  • Maple
    A047312:=n->(14*n-5+3*I^(2*n)-(3+3*I)*I^(-n)-(3-3*I)*I^n)/8: seq(A047312(n), n=1..100); # Wesley Ivan Hurt, Jun 03 2016
  • Mathematica
    Table[(14n-5+3*I^(2n)-(3+3*I)*I^(-n)-(3-3*I)*I^n)/8, {n, 80}] (* Wesley Ivan Hurt, Jun 03 2016 *)

Formula

G.f.: x^2*(4+x+x^2+x^3) / ( (1+x)*(1+x^2)*(x-1)^2 ). - R. J. Mathar, Oct 25 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-5+3*i^(2*n)-(3+3*i)*i^(-n)-(3-3*i)*i^n)/8 where i=sqrt(-1).
a(2k) = A047288(k), a(2k-1) = A047382(k). (End)

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
Showing 1-4 of 4 results.

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

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